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JEE Parabola PYQs with Solutions

411 previous year questions from Parabola (JEE Main & Advanced Mathematics) with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

411 questionsMathematics2003–2026Answers + solutions

About Parabola PYQs

Parabola has 411 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 220 questions in this list come from JEE Main (149), NTA Abhyas JEE Main (36) and JEE Advanced (35), spanning 2003 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

Questions by year (this list)

  • 2026: 16
  • 2025: 18
  • 2024: 13
  • 2023: 15
  • 2022: 20
  • 2021: 20
  • 2020: 47
  • 2019: 13
  • 2018: 6
  • 2017: 3
  • 2016: 3
  • 2015: 6

Parabola questions with solutions

1

Let T be the tangent to the parabola y^2 = 16x at the point (64, 32) . Let L be the tangent to the same parabola at another point (x_1, y_1) on the parabola. If L and T are

JEE Advanced 2026 Solution View → 2

Let O be the vertex of the parabola y^2=4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the

JEE Main 2026 Solution View → 3

Let chord PQ of length 3 13 of the parabola y^2 = 12x be such that the ordinates of points P and Q are in the ratio 1:2 . If the chord PQ subtends an angle α at the focus of the

JEE Main 2026 Solution View → 4

Let the point P be the vertex of the parabola y = x^2 - 6x + 12 . If a line passing through the point P intersects the circle x^2 + y^2 - 2x - 4y + 3 = 0 at the points R and S

JEE Main 2026 Solution View → 5

Let the directrix of the parabola P: y^2 = 8x , cut x -axis at the point A . Let B(α, β) , α > 1 , be a point on P such that the slope of AB is 3/5 . If BC is a focal chord of P

JEE Main 2026 Solution View → 6

Let A,B and C be the vertices of a variable right angled triangle inscribed in the parabola y^2=16x . Let the vertex B containing the right angle be (4,8) and the locus of the

JEE Main 2026 Solution View → 7

Consider the parabola P : y^2 = 4kx and the ellipse E : (x^2)/(a^2) + (y^2)/(b^2) = 1 . Let the line segment joining the points of intersection of P and E , be their latus

JEE Main 2026 Solution View → 8

Let the parabola y = x^2 + px + q passing through the point (1, -1) be such that the distance between its vertex and the x -axis is minimum. Then the value of p^2 + q^2 is:

JEE Main 2026 Solution View → 9

Let A be the focus of the parabola y^ 2 =8 x . Let the line y=m x+c intersect the parabola at two distinct points B and C . If the centroid of the triangle A B C is ((7)/(3)

JEE Main 2026 Solution View → 10

Let the image of parabola x^ 2 =4 y , in the line x-y=1 be (y+a)^ 2 =b(x-c) , a, b, c ∈ ~N . Then a+b+c is equal to

JEE Main 2026 Solution View → 11

An equilateral triangle OAB is inscribed in the parabola y^ 2 =4 x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having A B as a

JEE Main 2026 Solution View → 12

Let the locus of the mid-point of the chord through the origin O of the parabola y^ 2 =4 x be the curve S. Let P be any point on S. Then the locus of the point, which internally

JEE Main 2026 Solution View → 13

If the chord joining the points P_ 1 (x_ 1 , y_ 1 ) and P_ 2 (x_ 2 , y_ 2 ) on the parabola y^ 2 =12 x subtends a right angle at the vertex of the parabola, then x_ 1 x_ 2 -y_ 1

JEE Main 2026 Solution View → 14

Let y^ 2 =12 x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x -axis such that ∠ OPA=90^ ∘ . Then the locus of the centroid of

JEE Main 2026 Solution View → 15

Let one end of a focal chord of the parabola y^ 2 =16 x be (16,16) . If P(α, β) divides this focal chord internally in the ratio 5: 2 , then the minimum value of α+β is equal to :

JEE Main 2026 Solution View → 16

Let O be the vertex of the parabola x^ 2 =4 y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2: 3 be the conic

JEE Main 2026 Solution View → 17

Let S denote the locus of the mid-points of those chords of the parabola y^2=x , such that the area of the region enclosed between the parabola and the chord is (4)/(3) . Let R

JEE Advanced 2025 Solution View → 18

Let r be the radius of the circle, which touches x -axis at point (a, 0), a 0 and the parabola y^2=9 x at the point (4,6) . Then r is equal to ________

JEE Main 2025 Solution View → 19

Let P be the parabola, whose focus is (-2,1) and directrix is 2 x+y+2=0 . Then the sum of the ordinates of the points on P , whose abscissa is -2 , is

JEE Main 2025 Solution View → 20

The axis of a parabola is the line y=x and its vertex and focus are in the first quadrant at distances 2 and 2 2 units from the origin, respectively. If the point (1, k) lies on

JEE Main 2025 Solution View → 21

A line passing through the point A(-2,0) , touches the parabola P: y^2=x-2 at the point B in the first quadrant. The area, of the region bounded by the line AB , parabola P and

JEE Main 2025 Solution View → 22

The radius of the smallest circle which touches the parabolas y=x^2+2 and x=y^2+2 is

JEE Main 2025 Solution View → 23

Let the point P of the focal chord P Q of the parabola y^2=16 x be (1,-4) . If the focus of the parabola divides the chord PQ in the ratio m: n , gcd(m, n)=1 , then m^2+n^2 is

JEE Main 2025 Solution View → 24

Let the focal chord P Q of the parabola y^2=4 x make an angle of 60^ ∘ with the positive x -axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S

JEE Main 2025 Solution View → 25

Let y^2=12 x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ)=(147)/(4) . Let C be the circle described taking PQ as a diameter. If

JEE Main 2025 Solution View → 26

Two parabolas have the same focus \((4,3)\) and their directrices are the \(x\)-axis and the \(y\)-axis, respectively. If these parabolas intersects at the points \(A\) and \(B\)

JEE Main 2025 Solution View → 27

Let A and B be the two points of intersection of the line y+5=0 and the mirror image of the parabola y^2=4 x with respect to the line x+y+4=0 . If d denotes the distance between A

JEE Main 2025 Solution View → 28

Let A B C D be a trapezium whose vertices lie on the parabola y^2=4 x . Let the sides A D and B C of the trapezium be parallel to y -axis. If the diagonal AC is of length (25)/(4)

JEE Main 2025 Solution View → 29

If the equation of the parabola with vertex V((3)/(2), 3) and the directrix x+2 y=0 is α x^2+β y^2-γ x y-30 x-60 y+225=0 , then α+β+γ is equal to :

JEE Main 2025 Solution View → 30

Let the shortest distance from (a, 0), a 0 , to the parabola y^2=4 x be 4 . Then the equation of the circle passing through the point (a, 0) and the focus of the parabola, and

JEE Main 2025 Solution View → 31

The focus of the parabola y^2=4 x+16 is the centre of the circle C of radius 5 . If the values of λ , for which C passes through the point of intersection of the lines 3 x-y=0 and

JEE Main 2025 Solution View → 32

If the line 3 x-2 y+12=0 intersects the parabola 4 y=3 x^2 at the points A and B , then at the vertex of the parabola, the line segment A B subtends an angle equal to

JEE Main 2025 Solution View → 33

Let P(4,4 3 ) be a point on the parabola y^2=4 a x and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the

JEE Main 2025 Solution View → 34

Let the parabola y=x^2+p x-3 , meet the coordinate axes at the points P, Q and R . If the circle C with centre at (-1,-1) passes through the points P, Q and R , then the area of P

JEE Main 2025 Solution View → 35

Let A_1, B_1, C_1 be three points in the x y -plane. Suppose that the lines A_1 C_1 and B_1 C_1 are tangents to the curve y^2=8 x at A_1 and B_1 , respectively. If O=(0,0) and

JEE Advanced 2024 Solution View → 36

A normal with slope 1 6 is drawn from the point (0,-α) to the parabola x^2=-4 a y , where a>0 . Let L be the line passing through (0,-α) and parallel to the directrix of the

JEE Advanced 2024 Solution View → 37

Let A, B and C be three points on the parabola y^2=6 x and let the line segment A B meet the line L through C parallel to the x -axis at the point D . Let M and N respectively be

JEE Main 2024 Solution View → 38

Consider the circle C: x^2+y^2=4 and the parabola P: y^2=8 x . If the set of all values of α , for which three chords of the circle C on three distinct lines passing through the

JEE Main 2024 Solution View → 39

Let a conic C pass through the point (4,-2) and P(x, y), x ≥ 3 , be any point on C . Let the slope of the line touching the conic C only at a single point P be half the slope of

JEE Main 2024 Solution View → 40

Let L_1, L_2 be the lines passing through the point P(0,1) and touching the parabola 9 x^2+12 x+18 y-14=0 . Let Q and R be the points on the lines L_1 and L_2 such that the P Q R

JEE Main 2024 Solution View → 41

Let a line perpendicular to the line 2 x-y=10 touch the parabola y^2=4(x-9) at the point P . The distance of the point P from the centre of the circle x^2+y^2-14 x-8 y+56=0 is

JEE Main 2024 Solution View → 42

Suppose A B is a focal chord of the parabola y^2=12 x of length l and slope m < 3 . If the distance of the chord AB from the origin is d , then l ~d^2 is equal to _______

JEE Main 2024 Solution View → 43

Let PQ be a chord of the parabola y^2=12 x and the midpoint of PQ be at (4,1) . Then, which of the following point lies on the line passing through the points P and Q?

JEE Main 2024 Solution View → 44

Let the length of the focal chord PQ of the parabola y^2=12 x be 15 units. If the distance of PQ from the origin is p , then 10 p^2 is equal to _______

JEE Main 2024 Solution View → 45

The maximum area of a triangle whose one vertex is at ( 0, 0 ) and the other two vertices lie on the curve y = - 2 x ^ 2 + 54 at points ( x, y ) and ( - x, y ) where y > 0 is :

JEE Main 2024 Solution View → 46

Let P ( α, β ) be a point on the parabola y ^ 2 = 4x . If P also lies on the chord of the parabola x ^ 2 = 8y whose mid point is (1, (5)/(4)) , then ( α - 28 ) ( β - 8 ) is equal

JEE Main 2024 Solution View → 47

If the shortest distance of the parabola y ^ 2 = 4x from the centre of the circle x ^ 2 + y ^ 2 - 4x - 16y + 64 = 0 is d , then d ^ 2 is equal to :

JEE Main 2024 Solution View → 48

Let P be a point on the parabola y ^ 2 = 4ax , where a > 0 . The normal to the parabola at P meets the x -axis at a point Q . The area of the triangle PFQ , where F is the focus

JEE Advanced 2023 Solution View → 49

Let PQ be a focal chord of the parabola y ^ 2 = 36x of length 100, making an acute angle with the positive x - axis. Let the ordinate of P be positive and M be the point on the

JEE Main 2023 Solution View → 50

Let a common tangent to the curves y ^ 2 = 4x and (x - 4) ^ 2 + y ^ 2 = 16 touch the curves at the points P and Q . Then (PQ) ^ 2 is equal to ________.

JEE Main 2023 Solution View → 51

The ordinates of the points P and Q on the parabola with focus ( 3, 0 ) and directrix x = - 3 are in the ratio 3: 1 . If R ( α, β ) is the point of intersection of the tangents to

JEE Main 2023 Solution View → 52

Let R be the focus of the parabola y ^ 2 = 20x and the line y = mx + c intersect the parabola at two points P and Q . Let the points G(10, 10) be the centroid of the triangle PQR

JEE Main 2023 Solution View → 53

If the x -intercept of a focal chord of the parabola y ^ 2 = 8x + 4y + 4 is 3 , then the length of this chord is equal to _____ .

JEE Main 2023 Solution View → 54

Let S be the set of all a ∈ N such that the area of the triangle formed by the tangent at the point P(b, c), b, c ∈ N , on the parabola y ^ 2 = 2ax and the lines x = b, y = 0 is

JEE Main 2023 Solution View → 55

The parabolas : a x ^ 2 + 2bx + cy = 0 and d ^ 2 + 2ex + fy = 0 intersect on the line y = 1 . If a, b, c, d, e, f are positive real numbers and a, b, c are in G . P . , then

JEE Main 2023 Solution View → 56

Let A be a point on the x -axis. Common tangents are drawn from A to the curves x ^ 2 + y ^ 2 = 8 and y ^ 2 = 16x . If one of these tangents touches the two curves at Q and R

JEE Main 2023 Solution View → 57

If P ( h, k ) be point on the parabola x = 4 y ^ 2 , which is nearest to the point Q ( 0, 33 ) , then the distance of P from the directrix of the parabola y ^ 2 = 4 ( x + y ) is

JEE Main 2023 Solution View → 58

A triangle is formed by the tangents at the point ( 2, 2 ) on the curves y ^ 2 = 2x and x ^ 2 + y ^ 2 = 4x , and the line x + y + 2 = 0 . If r is the radius of its circumcircle

JEE Main 2023 Solution View → 59

The equations of two sides of a variable triangle are x = 0 and y = 3 , and its third side is a tangent to the parabola y ^ 2 = 6x . The locus of its circumcentre is :

JEE Main 2023 Solution View → 60

The distance of the point (6, - 2 2 ) from the common tangent y = mx + c, m > 0 , of the curves x = 2 y ^ 2 and x = 1 + y ^ 2 is

JEE Main 2023 Solution View → 61

The equations of sides AB and AC of a triangle ABC are (λ + 1)x + = 4 and + (1 - λ)y + λ = 0 respectively. Its vertex A is on the y - axis and its orthocentre is (1, 2) . The

JEE Main 2023 Solution View → 62

Let a tangent to the curve y ^ 2 = 24x meet the curve xy = 2 at the points A and B . Then the mid-points of such line segments AB lie on a parabola with the

JEE Main 2023 Solution View → 63

Consider the parabola y ^ 2 = 4x . Let S be the focus of the parabola. A pair of tangents drawn to the parabola from the point P = (- 2, 1) meet the parabola at P _ 1 and P _ 2 .

JEE Advanced 2022 Solution View → 64

Two tangent lines l _ 1 and l _ 2 are drawn from the point (2, 0) to the parabola 2 y ^ 2 = - x . If the lines l _ 1 and l _ 2 are also tangent to the circle (x - 5) ^ 2 + y ^ 2 =

JEE Main 2022 Solution View → 65

If the tangents drawn at the points P and Q on the parabola y ^ 2 = 2x - 3 intersect at the point R(0, 1) , then the orthocentre of the triangle PQR is

JEE Main 2022 Solution View → 66

If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a , is 16 , then |a| is equal to

JEE Main 2022 Solution View → 67

Let P(a, b) be a point on the parabola y ^ 2 = 8x such that the tangent at P passes through the centre of the circle x ^ 2 + y ^ 2 - 10x - 14y + 65 = 0 . Let A be the product of

JEE Main 2022 Solution View → 68

The equation of a common tangent to the parabolas y = x ^ 2 and y = - (x - 2) ^ 2 is

JEE Main 2022 Solution View → 69

The tangents at the points A(1, 3) and B(1, - 1) on the parabola y ^ 2 - 2x - 2y = 1 meet at the point P . Then the area (in unit ^ 2 ) of the triangle PAB is:

JEE Main 2022 Solution View → 70

The sum of diameters of the circles that touch (i) the parabola 75 x ^ 2 = 64(5y - 3) at the point ((8)/(5), (6)/(5)) and (ii) the y -axis, is equal to _____ .

JEE Main 2022 Solution View → 71

Let P: y ^ 2 = 4ax, a > 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of (π)/(4) with the line y = 3x + 5 touch the parabola P at A and B . Then

JEE Main 2022 Solution View → 72

If vertex of parabola is (2, - 1) and equation of its directrix is 4x - 3y = 21 , then the length of latus rectum is

JEE Main 2022 Solution View → 73

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y - 29 = 0 , is x ^ 2 + a y ^ 2 + bxy + cx + dy + k = 0 , then a + b + c + d + k is equal to

JEE Main 2022 Solution View → 74

A circle of radius 2 unit passes through the vertex and the focus of the parabola y ^ 2 = 2x and touches the parabola y = (x - (1)/(4)) ^ 2 + α , where α > 0 . Then (4α - 8) ^ 2

JEE Main 2022 Solution View → 75

Let the normal at the point P on the parabola y ^ 2 = 6x pass through the point (5, - 8) . If the tangent at P to the parabola intersects its directrix at the point Q , then the

JEE Main 2022 Solution View → 76

Let the common tangents to the curves 4( x ^ 2 + y ^ 2 ) = 9 and y ^ 2 = 4x intersect at the point Q . Let an ellipse, centered at the origin O , has lengths of semi-minor and

JEE Main 2022 Solution View → 77

If y = m _ 1 x + c _ 1 and y = m _ 2 x + c _ 2 , m _ 1 ≠ m _ 2 are two common tangents of circle x ^ 2 + y ^ 2 = 2 and parabola y ^ 2 = x , then the value of 8| m _ 1 m _ 2 | is

JEE Main 2022 Solution View → 78

Let x = 2t, y = t ^ 2 3 be a conic. Let S be the focus and B be the point on the axis of the conic such that SA ⊥ BA , where A is any point on the conic. If k is the ordinate of

JEE Main 2022 Solution View → 79

Let P _ 1 be a parabola with vertex (3, 2) and focus (4, 4) and P _ 2 be its mirror image with respect to the line x + 2y = 6 . Then the directrix of P _ 2 is x + 2y = _____.

JEE Main 2022 Solution View → 80

A particle is moving in the xy -plane along a curve C passing through the point (3, 3) . The tangent to the curve C at the point P meets the x -axis at Q . If the y -axis bisects

JEE Main 2022 Solution View → 81

Let x ^ 2 + y ^ 2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x ^ 2 at (2, 4) . Then A + C is equal to _____

JEE Main 2022 Solution View → 82

If two tangents drawn from a point (α, β) lying on the ellipse 25 x ^ 2 + 4 y ^ 2 = 1 to the parabola y ^ 2 = 4x are such that the slope of one tangent is four times the other

JEE Main 2022 Solution View → 83

Let E denote the parabola y ^ 2 = 8x . Let P = (- 2, 4) , and let Q and Q ^ ' be two distinct points on E such that the lines PQ and P Q ^ ' are tangents to E . Let F be the focus

JEE Advanced 2021 Solution View → 84

Consider the region R=\ (x, y) ∈ ℝ × ℝ: x ≥ 0. and .y^ 2 ≤ 4-x\ . Let F be the family of all circles that are contained in R and have centers on the x -axis. Let C be the circle

JEE Advanced 2021 Solution View → 85

Consider the region R=\ (x, y) ∈ ℝ × ℝ: x ≥ 0. and .y^ 2 ≤ 4-x\ . Let F be the family of all circles that are contained in R and have centers on the x -axis. Let C be the circle

JEE Advanced 2021 Solution View → 86

Consider the parabola with vertex ((1)/(2), (3)/(4)) and the directrix y = (1)/(2) . Let P be the point where the parabola meets the line x = - (1)/(2) . If the normal to the

JEE Main 2021 Solution View → 87

A tangent line L is drawn at the point (2, - 4) on the parabola y ^ 2 = 8x . If the line L is also tangent to the circle x ^ 2 + y ^ 2 = a, then a is equal to .

JEE Main 2021 Solution View → 88

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x -axis at a distance R and S ( > R ) respectively from the origin, is :

JEE Main 2021 Solution View → 89

If two tangents drawn from a point P to the parabola y ^ 2 = 16 ( x - 3 ) are at right angles, then the locus of point P is:

JEE Main 2021 Solution View → 90

A tangent and a normal are drawn at the point P ( 2, - 4 ) on the parabola y ^ 2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q ( a, b )

JEE Main 2021 Solution View → 91

If a line along a chord of the circle 4 x ^ 2 + 4 y ^ 2 + 120x + 675 = 0 , passes through the point ( - 30, 0 ) and is tangent to the parabola y ^ 2 = 30x , then the length of

JEE Main 2021 Solution View → 92

Let a parabola P be such that its vertex and focus lie on the positive x -axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O ( 0, 0 ) to

JEE Main 2021 Solution View → 93

Let P be a variable point on the parabola y = 4 x ^ 2 + 1 . Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y

JEE Main 2021 Solution View → 94

If the point on the curve y ^ 2 = 6x, nearest to the point (3, (3)/(2)) is ( α, β ) , then 2 ( α + β ) is equal to _________.

JEE Main 2021 Solution View → 95

Let the tangent to the parabola S: y ^ 2 = 2x at the point P(2, 2) meet the x - axis at Q and normal at it meet the parabola S at the point R . Then the area (in sq. units) of the

JEE Main 2021 Solution View → 96

Let y = mx + c, m > 0 be the focal chord of y ^ 2 = - 64x , which is tangent to ( x + 10 ) ^ 2 + y ^ 2 = 4 . Then, the value of 4 2 ( m + c ) is equal to______

JEE Main 2021 Solution View → 97

Let C be the locus of the mirror image of a point on the parabola y ^ 2 = 4x with respect to the line y = x . Then the equation of tangent to C at P(2, 1) is :

JEE Main 2021 Solution View → 98

If the three normals drawn to the parabola, y ^ 2 = 2x pass through the point (a, 0), a ≠ 0 , then a must be greater than :

JEE Main 2021 Solution View → 99

A line is a common tangent to the circle (x - 3) ^ 2 + y ^ 2 = 9 and the parabola y ^ 2 = 4x . If the two points of contact (a, b) and (c, d) are distinct and lie in the first

JEE Main 2021 Solution View → 100

A tangent is drawn to the parabola y ^ 2 = 6x which is perpendicular to the line 2x + y = 1 . Which of the following points does NOT lie on it?

JEE Main 2021 Solution View → 101

If P is a point on the parabola y = x ^ 2 + 4 which is closest to the straight line y = 4x - 1, then the co-ordinates of P are:

JEE Main 2021 Solution View → 102

The locus of the mid-point of the line segment joining the focus of the parabola y ^ 2 = 4ax to a moving point of the parabola, is another parabola whose directrix is:

JEE Main 2021 Solution View → 103

The centre of the circle passing through the point (0, 1) and touching the parabola y = x ^ 2 at the point (2, 4) is :

JEE Main 2020 Solution View → 104

Let L _ 1 be a tangent to the parabola y ^ 2 = 4 ( x + 1 ) and L _ 2 be a tangent to the parabola y ^ 2 = 8 ( x + 2 ) such that L _ 1 and L _ 2 intersect at right angles. Then L _

JEE Main 2020 Solution View → 105

If the common tangent to the parabolas, y ^ 2 = 4xand x ^ 2 = 4y also touches the circle, x ^ 2 + y ^ 2 = c ^ 2 , then c is equal to :

JEE Main 2020 Solution View → 106

Let the latus rectum of the parabola y ^ 2 = 4x be the common chord to the circles C _ 1 and C _ 2 each of them having radius 2 5 . Then, the distance between the centres of the

JEE Main 2020 Solution View → 107

Let P be a point on the parabola, y ^ 2 = 12x and N be the foot of the perpendicular drawn from P , on the axis of the parabola. A line is now drawn through the mid-point M of PN

JEE Main 2020 Solution View → 108

The area (in sq. units) of an equilateral triangle inscribed in the parabola y ^ 2 = 8x, with one of its vertices on the vertex of this parabola is

JEE Main 2020 Solution View → 109

If one end of a focal chord AB of the parabola y ^ 2 = 8x is at A((1)/(2), - 2), then the equation of the tangent to it at B is:

JEE Main 2020 Solution View → 110

Let a line y = mx (m > 0), intersect the parabola, y ^ 2 = x , at a point P, other than the origin. Let the tangent to it a P , meet the x -axis at the point Q . If area ( ) = 4

JEE Main 2020 Solution View → 111

The locus of a point which divides the line segment joining the point (0, - 1) and a point on the parabola x ^ 2 = 4y internally in the ratio 1: 2 is:

JEE Main 2020 Solution View → 112

For a > 0, let the curves C _ 1 : y ^ 2 = ax and C _ 2 : x ^ 2 = ay intersect at origin O and a point P . Let the line x = b(0 < b < a) intersect the chord OP and the x -axis at

JEE Main 2020 Solution View → 113

If y = mx + 4 is a tangent to both the parabolas, y ^ 2 = 4x and x ^ 2 = 2by, then b is equal to

JEE Main 2020 Solution View → 114

P is a point on the parabola whose ordinate equals its abscissa. A normal is drawn to the parabola at P to meet it again at Q . If S is the focus of the parabola, then the product

NTA Abhyas JEE Main 2020 Solution View → 115

If the line y - 2 = 0 is the directrix of the parabola x ^ 2 - ky + 32 = 0, k ≠ 0 and the parabola intersects the circle x ^ 2 + y ^ 2 = 8 at two real distinct points, then the

NTA Abhyas JEE Main 2020 Solution View → 116

The coordinates of the focus of the parabola described parametrically by x = 5t^ 2 + 2, y = 10t + 4 are

NTA Abhyas JEE Main 2020 Solution View → 117

The length of the chord of the parabola x^ 2 = 4y having equation x - 2 y + 4 2 = 0 is

NTA Abhyas JEE Main 2020 Solution View → 118

The equation of the common tangent to the curves y ^ 2 = 4x and x ^ 2 + 32y = 0 is x + by + c = 0 . The value of | ^ – 1 ( sin1 ) + ^ – 1 ( sinb ) + ^ – 1 ( sinc ) | is equal to

NTA Abhyas JEE Main 2020 Solution View → 119

The locus of the middle points of the chords of the parabola y^ 2 = 4ax, which passes through the origin is

NTA Abhyas JEE Main 2020 Solution View → 120

If two parabolas y^ 2 = 4a (x – k) and x^ 2 = 4a (y – k) have only one common point P , then the equation of normal to y^ 2 = 4a (x – k) at P is

NTA Abhyas JEE Main 2020 Solution View → 121

The equation of parabola which cuts the parabola y^ 2 = 4bx orthogonally having the same axis and length of latus rectum as 8b , is

NTA Abhyas JEE Main 2020 Solution View → 122

The parabolas C _ 1 : y ^ 2 = 4a(x - a) and C _ 2 : y ^ 2 = - 4a(x - k) intersect at two distinct points A and B . If the slope of the tangent at A on C _ 1 is same as the slope

NTA Abhyas JEE Main 2020 Solution View → 123

If the line y = x - 1 bisects two chords of the parabola y^ 2 = 4bx which are passing through the point~ (b, - 2b), then the length of the latus rectum can be equal to

NTA Abhyas JEE Main 2020 Solution View → 124

Let the lines l _ 1 and l _ 2 be normals to y^ 2 = 4x and tangents to x^ 2 = - 12y (where l _ 1 and l _ 2 are not x -axis). The absolute value of the difference of slopes of l _ 1

NTA Abhyas JEE Main 2020 Solution View → 125

If two tangents drawn from the point P ( h, k ) to the parabola y^ 2 = 8x are such that the slope of one of the tangent is 3 times the slope of the other, then the locus of point

NTA Abhyas JEE Main 2020 Solution View → 126

If the equation of the tangent at the point P(3,4) on the parabola whose axis is the x -axis is 3x - 4y + 7 = 0, then the distance of the tangent from the focus of the parabola is

NTA Abhyas JEE Main 2020 Solution View → 127

Let the focus S of the parabola y^ 2 =8 x lies on the focal chord PQ of the same parabola. If PS = 6, then the square of the slope of the chord PQ & is

NTA Abhyas JEE Main 2020 Solution View → 128

The equation of the tangent to the parabola y^ 2 = 4x whose slope is positive and which also touches x^ 2 + y^ 2 = (1)/(2) is

NTA Abhyas JEE Main 2020 Solution View → 129

Tangents to the parabola& y^ 2 = 4ax at P (a t _ 1 ^ 2 , 2a t _ 1 ) and Q (a t _ 2 ^ 2 , 2a t _ 2 ) meet at T . If △ PTQ is right-angled at T, then (1)/(PS) + (1)/(QS) is equal to

NTA Abhyas JEE Main 2020 Solution View → 130

If ( 9a, 6a ) is a point bounded in the region formed by parabola y^ 2 = 16x and x = 9, then

NTA Abhyas JEE Main 2020 Solution View → 131

The length of the focal chord of the parabola y^ 2 = 4x at a distance of 0.4 units from the origin is

NTA Abhyas JEE Main 2020 Solution View → 132

Let points A _ 1 , A _ 2 ~and& A _ 3 lie on the parabola y^ 2 = 8x . If △ A _ 1 A _ 2 A _ 3 is an equilateral triangle and normals at points A _ 1 , A _ 2 ~and& A _ 3 on this

NTA Abhyas JEE Main 2020 Solution View → 133

Let PQ be the focal chord of the parabola y ^ 2 = 4x. If the centre of the circle having PQ as its diameter lies on the line y = - 4 5 , then the length (in units) of chord PQ is

NTA Abhyas JEE Main 2020 Solution View → 134

Two tangents are drawn from a point ( – 4, 3 ) to the parabola y^ 2 = 16x . If α is the angle between them, then the value of cos α is

NTA Abhyas JEE Main 2020 Solution View → 135

The set of points on the axis of the parabola y^ 2 - 2y - 4x + 5 = 0 from which all the three normals drawn to the parabola are real and distinct, is

NTA Abhyas JEE Main 2020 Solution View → 136

Let lx - 2y = 1 intersects the parabola y ^ 2 = 4ax at points P and Q . If PS and QS meet the parabola again at R and T respectively (where S is the focus of y ^ 2 = 4ax ), then

NTA Abhyas JEE Main 2020 Solution View → 137

The radius of circle, touching the parabola y^ 2 = 8xat (2,4) and passing through (0,4), is

NTA Abhyas JEE Main 2020 Solution View → 138

Let normals to the parabola y ^ 2 = 4x at variable points P( t _ 1 ^ 2 , 2 t _ 1 ) and Q( t _ 2 ^ 2 , 2 t _ 2 ) meet at the point R( t ^ 2 , 2t), then the line joining P and Q

NTA Abhyas JEE Main 2020 Solution View → 139

Let the tangents at points& P and R on the parabola y = x^ 2 intersects at T . Tangent at point Q (lies in between the points P and R ) on the parabola intersect PT and RT at A

NTA Abhyas JEE Main 2020 Solution View → 140

If the normals at two points P and Q of a parabola y ^ 2 = 4x intersect at a third point R on the parabola y ^ 2 = 4x , then the product of the ordinates of P and Q is equal to

NTA Abhyas JEE Main 2020 Solution View → 141

The length of the chord y = 3 x - 2 3 intercepted by the parabola y^ 2 = 4 (x - 1) is equal to

NTA Abhyas JEE Main 2020 Solution View → 142

Let the normals at points A (4a, - 4a) and B (9a, - 6a) on the parabola y^ 2 = 4ax meet at the point P. The equation of the normal from P on y^ 2 = 4ax (other than PA and PB )& is

NTA Abhyas JEE Main 2020 Solution View → 143

OA is the chord of the parabola y^ 2 = 4x (where O is the origin). AB is a chord of y^ 2 = 4x and perpendicular to OA which cuts the axis of the parabola at C. If the foot of A on

NTA Abhyas JEE Main 2020 Solution View → 144

Chord joining two distinct points P (a, 4b) and Q (c, - (16)/(b)) (both are variable points) on the parabola y^ 2 = 16x always passes through a fixed point (α, β) .Then, which of

NTA Abhyas JEE Main 2020 Solution View → 145

If three normals are drawn from the point& (c, 0) to the parabola~ y^ 2 = 4x and two of which are perpendicular, then the value of~ c is equal to

NTA Abhyas JEE Main 2020 Solution View → 146

Let tangents PQ and PR are drawn from the point P(- 2,4) to the parabola y ^ 2 = 4x . If S is the focus of the parabola y ^ 2 = 4x , then the value (in units) of RS + SQ is equal

NTA Abhyas JEE Main 2020 Solution View → 147

If the line y = - (7)/(2) is the directrix of the parabola x ^ 2 - ky + 8 = 0 , then the sum of all the possible values of k is equal to

NTA Abhyas JEE Main 2020 Solution View → 148

The equation of the tangent to the parabola y^ 2 = 4x + 5 which is parallel to the line y = 2x + 7 is

NTA Abhyas JEE Main 2020 Solution View → 149

Let the tangents PQ and PR are drawn to y^ 2 = 4ax from any point P on the line x + 4a = 0. The angle subtended by the chord of contact QR at the vertex of the parabola y^ 2 = 4ax

NTA Abhyas JEE Main 2020 Solution View → 150

The tangents to the curve y = (x - 2) ^ 2 - 1 at its points of intersection with the line x - y = 3, intersect at the point:

JEE Main 2019 Solution View → 151

If the line ax + y = c , touches both the curves x ^ 2 + y ^ 2 = 1 and y ^ 2 = 4 2 x , then |c| is equal to:

JEE Main 2019 Solution View → 152

The area (in sq. units) of the smaller of the two circles that touch the parabola, y ^ 2 = 4x at the point (1, 2) and the x -axis is

JEE Main 2019 Solution View → 153

If one end of a focal chord of the parabola, y ^ 2 = 16x is at (1, 4) , then the length of this focal chord is

JEE Main 2019 Solution View → 154

The tangent to the parabola y ^ 2 = 4x at the point where it intersects the circle x ^ 2 + y ^ 2 = 5 in the first quadrant, passes through the point:

JEE Main 2019 Solution View → 155

The equation of a tangent to the parabola, x ^ 2 = 8y , which makes an angle θ with the positive direction of x - axis, is

JEE Main 2019 Solution View → 156

Let P(4, - 4) and Q(9, 6) be two points on the parabola, y ^ 2 = 4x and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the

JEE Main 2019 Solution View → 157

If the area of the triangle whose one vertex is at the vertex of the parabola, y^ 2 +4(x-a^ 2 )=0 and the other two vertices are the points of intersection of the parabola and y

JEE Main 2019 Solution View → 158

The length of the chord of the parabola x ^ 2 = 4y having equation x - 2 y + 4 2 = 0 is

JEE Main 2019 Solution View → 159

If the parabolas y ^ 2 = 4b (x - c) and y ^ 2 = 8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)

JEE Main 2019 Solution View → 160

Let A(4, - 4) and B(9, 6) be points on the parabola, y ^ 2 = 4x . Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of is maximum. Then

JEE Main 2019 Solution View → 161

Axis of a parabola lies along x -axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x -axis then which of following points does

JEE Main 2019 Solution View → 162

Equation of a common tangent to the circle, x ^ 2 + y ^ 2 - 6x = 0 and the parabola, y ^ 2 = 4x is:

JEE Main 2019 Solution View → 163

Let P be a point on the parabola x ^ 2 = 4y . If the distance of P from the center of the circle x ^ 2 + y ^ 2 + 6x + 8 = 0 is minimum, then the equation of the tangent to the

JEE Main 2018 Solution View → 164

Tangents drawn from the point (-8,0) to the parabola y^2=8 x touch the parabola at P and Q . If F is the focus of the parabola, then the area of the triangle P F Q (in sq. units)

JEE Main 2018 Solution View → 165

Two parabolas with a common vertex and with axes along x -axis and y -axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each

JEE Main 2018 Solution View → 166

Two parabolas with a common vertex and with axes along the x -axis and y -axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each

JEE Main 2018 Solution View → 167

If the tangent at (1, 7) to the curve x ^ 2 = y - 6 touch the circle x ^ 2 + y ^ 2 + 16x + 12y + c = 0 then the value of c is:

JEE Main 2018 Solution View → 168

Tangent and normal are drawn at P(16, 16) on the parabola y ^ 2 = 16x , which intersect the axis of the parabola at A and B , respectively. If C is the center of the circle

JEE Main 2018 Solution View → 169

If a chord, which is not a tangent, of the parabola y ^ 2 = 16x has the equation 2x + y = p , and midpoint (h, k), then which of the following is (are) possible value(s) of p, h

JEE Advanced 2017 Solution View → 170

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. Column 1 Column 2 Column 3 (I) x ^ 2 + y ^ 2 = a ^ 2 (i) my = m ^ 2 x +

JEE Advanced 2017 Solution View → 171

If y = mx + c is the normal at a point on the parabola y ^ 2 = 8x whose focal distance is 8 units, then |c| is equal to:

JEE Main 2017 Solution View → 172

Let P be the point on the parabola y ^ 2 = 4x which is at the shortest distance from the center S of the circle x ^ 2 + y ^ 2 - 4x - 16y + 64 = 0 . Let Q be the point on the

JEE Advanced 2016 Solution View → 173

P and Q are two distinct points on the parabola, y ^ 2 = 4x , with parameters t and t _ 1 , respectively. If the normal at P passes through Q , then the minimum value of t _ 1 ^ 2

JEE Main 2016 Solution View → 174

Let P be the point on the parabola, y ^ 2 = 8x which is at a minimum distance from the center C of the circle x ^ 2 + (y + 6) ^ 2 = 1 . Then the equation of the circle, passing

JEE Main 2016 Solution View → 175

Suppose that the foci of the ellipse x ^ 2 9 + y ^ 2 5 = 1 are ( f _ 1 , 0)and ( f _ 2 , 0 ) where f _ 1 > 0and f _ 2 < 0 . Let P _ 1 and P _ 2 be two parabolas with a common

JEE Advanced 2015 Solution View → 176

If the normal of the parabola y ^ 2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x - 3) ^ 2 + (y + 2) ^ 2 = r ^ 2 , then the value of r ^ 2 is

JEE Advanced 2015 Solution View → 177

Let P and Q be distinct points on the parabola y ^ 2 = 2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and

JEE Advanced 2015 Solution View → 178

If PQ be a double ordinate of the parabola, y ^ 2 = - 4x, where P lies in the second quadrant. If R divides PQ in the ratio 2: 1, then the locus of R is:

JEE Main 2015 Solution View → 179

If the tangent to the conic, y - 6 = x ^ 2 at ( 2, 10 ) touches the circle, x ^ 2 + y ^ 2 + 8x - 2y = k (for some fixed k ) at a point (α, β); then ( α, β ) is

JEE Main 2015 Solution View → 180

Let O be the vertex and Q be any point on the parabola, x ^ 2 = 8y . If the point P divides the line segment OQ internally in the ratio 1: 3 , then the locus of P is

JEE Main 2015 Solution View → 181

The common tangents to the circle x ^ 2 + y ^ 2 = 2 and the parabola y ^ 2 = 8x touch the circle at the points P, and Q the parabola at the points R, S. Then the area of the

JEE Advanced 2014 Solution View → 182

Paragraph: Let a, r, s, t be nonzero real numbers. Let P(a t^ 2 , 2 a t), Q, R(a r^ 2 , 2 a r) and S(a s^ 2 , 2 a s) be distinct points on the parabola y^ 2 =4 a x . Suppose that

JEE Advanced 2014 Solution View → 183

Paragraph: Let a, r, s, t be nonzero real numbers. Let P(a t^ 2 , 2 a t), Q, R(a r^ 2 , 2 a r) and S(a s^ 2 , 2 a s) be distinct points on the parabola y^ 2 =4 a x . Suppose that

JEE Advanced 2014 Solution View → 184

A chord is drawn through the focus of the parabola y ⁡ ^ 2 = 6 x ⁡ such that its distance from the vertex of this parabola is 5 2 , then its slope can be

JEE Main 2014 Solution View → 185

Two tangents are drawn from a point (-2,-1) to the curve, y^2=4 x . If α is the angle between them, then | α| is equal to:

JEE Main 2014 Solution View → 186

The locus of the foot of perpendicular drawn from the centre of the ellipse x ^ 2 + 3 y ^ 2 = 6 on any tangent to it is

JEE Main 2014 Solution View → 187

Paragraph: Let P Q be a focal chord of the parabola y^ 2 =4 a x . The tangents to the parabola at P and Q meet at a point lying on the line y=2 x+a, a>0 . Question: Length of

JEE Advanced 2013 Solution View → 188

Paragraph: Let P Q be a focal chord of the parabola y^ 2 =4 a x . The tangents to the parabola at P and Q meet at a point lying on the line y=2 x+a, a>0 . Question: If chord P Q

JEE Advanced 2013 Solution View → 189

The point of intersection of the normals to the parabola y^2=4 x at the ends of its latus rectum is :

JEE Main 2013 Solution View → 190

Statement-1: The line x-2 y=2 meets the parabola, y^2+2 x=0 only at the point (-2,-2) . Statement-2: The line y=m x-(1)/(2 m)(m ≠ 0) is tangent to the parabola, y^2=-2 x at the

JEE Main 2013 Solution View → 191

Statement-1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point

JEE Main 2013 Solution View → 192

Let S be the focus of the parabola y^ 2 =8 x and let P Q be the common chord of the circle x^ 2 +y^ 2 -2 x-4 y=0 and the given parabola. The area of the triangle P Q S is

JEE Advanced 2012 Solution View → 193

The chord P Q of the parabola y^2=x , where one end P of the chord is at point (4,-2) , is perpendicular to the axis of the parabola. Then the slope of the normal at Q is

JEE Main 2012 Solution View → 194

The equation of the normal to the parabola, x^2=8 y at x=4 is

JEE Main 2012 Solution View → 195

The area of triangle formed by the lines joining the vertex of the parabola, x^2=8 y , to the extremities of its latus rectum is

JEE Main 2012 Solution View → 196

Statement 1: y=m x-(1)/(m) is always a tangent to the parabola, y^2=-4 x for all non-zero values of m . Statement 2: Every tangent to the parabola, y^2=-4 x will meet its axis at

JEE Main 2012 Solution View → 197

Statement 1 : An equation of a common tangent to the parabola y^2=16 3 x and the ellipse 2 x^2+y^2=4 is y=2 x+2 3 . Statement 2 : If the line y=m x+ 4 3 m ,(m ≠ 0) is a common

JEE Main 2012 Solution View → 198

Let (x, y) be any point on the parabola y^2=4 x . Let P be the point that divides the line segment from (0,0) to (x, y) in the ratio 1: 3 . Then, the locus of P is

JEE Advanced 2011 Solution View → 199

Let L be a normal to the parabola y^2=4 x . If L passes through the point (9,6) , then L is given by

JEE Advanced 2011 Solution View → 200

Consider the parabola y^2=8 x . Let Δ_1 be the area of the triangle formed by the end points of its latusrectum and the point P((1)/(2), 2) on the parabola and Δ_2 be the area of

JEE Advanced 2011 Solution View →
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Related JEE Mathematics chapters

Application of Derivatives (795)Three Dimensional Geometry (756)Sequences and Series (748)Functions (700)Definite Integration (692)Vector Algebra (616)Differential Equations (586)Continuity and Differentiability (576)Probability (561)Complex Number (551)Binomial Theorem (550)Quadratic Equation (517)

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BITSAT Parabola (17)