Q: 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 denote the region lying in t

aligned & T=S_1 \\ & k y- ( x+h 2 )=k^2-h \\ & x-2 k y+2 k^2-h=0 \\ & k^2-h 0 \\ & For Area : interchange x \& y \\ & y-2 k x+2 k^2-h=0 \& y=x^2 \\ & x^2-2 k x+ (2 k^2-h )=0 h - k ^2=1 (4, 3 ) S aligned & A = _1^4( x - x -1 ) dx = 2 3 ( x ^ 3 / 2 -( x -1)^ 3 / 2 )_1^4 \\ & ~A = 2 3 (8-3 3 -1)= 2 3 (7-3 3 )= 14 3 -2 3 aligned

Q: Let E denote the parabola y 2 = 8 x . Let P = - 2 , 4 , and let Q and Q ' be two distinct points on E such that the lines P Q &n

y 2 = 8 x ⇒ a = 2 , Focus F 2 , 0 , and Directrix x = − 2 ∵ P − 2 , 4 lies on directrix ⇒ Q Q ' will be a focal chord and tangents are mutually perpendicular. ⇒ Δ Q P Q ' is right angled. Also the portion between point of contact and directrix of parabola subtends right angle at the focus. So, △ P F Q is a right- angled triangle. P F = − 2 −

Q: 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 that has largest radius amo

equation of normal at P is y - β = 2 β x - 4 + β 2 Passing through ( r , 0 ) ⇒ - 1 = 2 r - 4 + β 2 . . . i or β = 0 . . . i i Also r 2 = r + β 2 - 4 2 + β 2 . . . i i i from i and i i i r = 3 2 , - 5 2 so radius of circle is 1 . 5 and α = 2 from i i and i i i r = 2 but in this case circle intersect at three points

Q: 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 that has largest radius amo

equation of normal at P is y - β = 2 β x - 4 + β 2 Passing through ( r , 0 ) ⇒ - 1 = 2 r - 4 + β 2 . . . i or β = 0 . . . i i Also r 2 = r + β 2 - 4 2 + β 2 . . . i i i from i and i i i r = 3 2 , - 5 2 so radius of circle is 1 . 5 and α = 2 from i i and i i i r = 2 but in this case circle intersect at three points

Question 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 perpendicular to each other,

Accepted Answer

The equation of the parabola is y^2 = 16x. Comparing with y^2 = 4ax, we get a = 4. Let the parametric coordinates of the point (64, 32) be (at_1^2, 2at_1). 2at_1 = 32 8t_1 = 32 t_1 = 4 The slope of the tangent T at t_1 is m_1 = 1 t_1 = 1 4 . Let the tangent L be at the point (x_1, y_1) with parameter t_2. Its slope is m_2 = 1 t_2 . Since T and L are perpendicular, m_1 m_2 = -1. 1 4 1 t_2 = -1 t_2 = - 1 4 The x-coordinate of the point (x_1, y_1) is x_1 = at_2^2 = 4 (- 1 4 )^2 = 1 4 . The distance

Question 2

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 denote the region lying in t

Accepted Answer

aligned & T=S_1 \ & k y- ( x+h 2 )=k^2-h \ & x-2 k y+2 k^2-h=0 \ & k^2-h 0 \ & For Area : interchange x \& y \ & y-2 k x+2 k^2-h=0 \& y=x^2 \ & x^2-2 k x+ (2 k^2-h )=0 h - k ^2=1 (4, 3 ) S aligned & A = _1^4( x - x -1 ) dx = 2 3 ( x ^ 3 / 2 -( x -1)^ 3 / 2 )_1^4 \ & ~A = 2 3 (8-3 3 -1)= 2 3 (7-3 3 )= 14 3 -2 3 aligned

Question 3

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 C_1=(-4,0), then which of

Accepted Answer

Equation of tangent at (2 t^2, 4 t ) is ty = x +2 t ^2 It is passing through (-4,0) 0=-4+2 t ^2 t = 2 . array rl A _1=(4,4 2 ) \ B _1=(4,-4 2 ) array \ aligned & OA _1= 48 =4 3 \ & ~A _1 ~B _1=8 2 aligned Equation of altitude of A_1 B_1 C_1 drawn from A_1 is y-4 2 = 2 (x-4) 2 x-y=0 ...(1) Equation of altitude of A_1 B_1 C_1 drawn from C_1 is x =0 ...(2) Solving (1) and (2) orthocentre is (0,0) correct options are (1), (3)

Question 4

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

Accepted Answer

d y d x = . x -2 a d y d x |_N=-t Slope of normal = 1 t = 1 6 t= 6 Now, - at ^2+ 2 at = 1 t - at ^2+ =2 a -6 a + =2 a =8 a For A and B aligned & x^2=-4 a(-8 a) \ & x^2=32 a^2 x= 4 2 a \ & A(-4 2 a,-8 a), B(4 2 a,-8 a) \ & A B^2=(8 2 a)^2=128 a^2=s aligned Length of LR = r =4 a aligned & r s = 4 a 128 a ^2 = 1 16 \ & 32 a =16 a = 1 2 \ & 24 a =12 Ans. aligned

Question 5

Let  P  be a point on the parabola y 2 = 4 a x , where a &#62; 0 . The normal to the parabola at P meets the x -axis at a point Q . The area of the triangle P F Q , where  F is the focu

Accepted Answer

Given, Parabola  y 2 = 4 a x , Now plotting the normal to parabola at point P a m 2 , - 2 a m , we get So, equation of normal at  P a m 2 , &#160; - 2 a m  is  y = m x - 2 a m - a m 3 Hence, point  Q 2 a + a m 2 , 0 And focus of parabola will be  F a , 0 Now finding the area of  &#8710; P F Q  we get, Area of  &#8710; P F Q = 1 2 a + a m 2 &#215; 2 a m = 120 &#8658; a 2 m 1 + m 2 = 120 So, possible pair from option will be, a , &#160; m &#8801; 2 , &#

Question 6

Consider the parabola y 2 = 4 x . 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 . Let Q 1  a

Accepted Answer

On plotting the diagram of the given value we have, Now let  P 1 t 2 , 2 t  then tangent at  P 1  will be  t y = x + t 2 Since it passes through  - 2 , 1 &#8658; &#160; t 2 - t - 2 = 0 &#8658; t = 2 , - 1 So, we get points  P 1 4 , 4  and  P 2 1 , - 2 Now finding the equation of  S P 1 : 4 x - 3 y - 4 = 0 And equation of  S P 2 : x - 1 = 0 Now finding the point by foot of point on line formula, Q 1 : x 1 + 2 4 = y 1 - 1 - 3 = - - 8 - 3 - 4 2

Question 7

Let  E  denote the parabola  y 2 = 8 x .  Let  P = - 2 , 4 , and let  Q  and  Q '  be two distinct points on  E  such that the lines  P Q &n

Accepted Answer

y 2 = 8 x &#8658; a = 2 ,  Focus  F 2 , 0 ,  and Directrix  &#160; x = &#8722; 2 &#8757; &#160; P &#8722; 2 , 4  lies on directrix &#8658; &#160; Q Q '  will be a focal chord and tangents are mutually perpendicular. &#8658; &#160; &#160; &#916; Q P Q '  is right angled. Also the portion between point of contact and directrix of parabola subtends right angle at the focus. So,  &#9651; P F Q  is a right- angled triangle. P F = &#8722; 2 &#8722;

Question 8

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 that has largest radius amo

Accepted Answer

equation of normal at P  is y - &#946; = 2 &#946; x - 4 + &#946; 2 Passing through ( r , 0 ) &#8658; - 1 = 2 r - 4 + &#946; 2 &#160; . . . i or &#160;&#160; &#946; = 0 &#160; . . . i i Also r 2 = r + &#946; 2 - 4 2 + &#946; 2 &#160; . . . i i i from i  and i i i r = 3 2 , - 5 2 so radius of circle is 1 . 5 and &#945; = 2 from i i and i i i r = 2 but in this case circle intersect at three points

Question 9

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 that has largest radius amo

Accepted Answer

equation of normal at P  is y - &#946; = 2 &#946; x - 4 + &#946; 2 Passing through ( r , 0 ) &#8658; - 1 = 2 r - 4 + &#946; 2 &#160; . . . i or &#160;&#160; &#946; = 0 &#160; . . . i i Also r 2 = r + &#946; 2 - 4 2 + &#946; 2 &#160; . . . i i i from i  and i i i r = 3 2 , - 5 2 so radius of circle is 1 . 5 and &#945; = 2 from i i and i i i r = 2 but in this case circle intersect at three points

Parabola — Previous Year Questions & Solutions