Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced 2016Paper 2
Exam
JEE Advanced
Year
2016
Paper
Paper 2
Subject
Mathematics
Chapter
Parabola

JEE Advanced 2016 · Paper 2 · Mathematics · Parabola

Let P be the point on the parabola y ^ 2 = 4x which is at the...

Let P be the point on the parabola y2=4x{y}^{2} = 4x which is at the shortest distance from the center S of the circle x2+y2−4x−16y+64=0.{x}^{2} + {y}^{2} - 4x - 16y + 64 = 0 . Let Q be the point on the circle dividing the line segment SP internally. Then -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 +

Options

  1. ASP=25SP = 2\sqrt{5}
  2. BSQ:QP=(5+1):2SQ: QP = \left(\sqrt{5} + 1\right): 2
  3. CThe x - intercept of the normal to the parabola at P is 6
  4. DThe slope of the tangent to the circle at Q is 12\dfrac{1}{2}

Correct answer

A. SP=25SP = 2\sqrt{5}

Step-by-step solution

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 +
y2=4x{y}^{2} = 4x
Point P lies on normal to parabola passing through centre of circle
y+tx=2t+t3y + tx = 2t + {t}^{3} ......(i)
8+2t=2t+t38 + 2t = 2t + {t}^{3}
t=2t = 2
P(4,4)P\left(4, 4\right)
SP=(4−2)2+(4−8)2SP = \sqrt{{\left(4 - 2\right)}^{2} + {\left(4 - 8\right)}^{2}}
SP=25SP = 2\sqrt{5}
SQ=2SQ = 2
⇒⇒ PQ=25−2PQ = 2\sqrt{5} - 2
SQQP=15−1=5+14\dfrac{SQ}{QP} = \dfrac{1}{\sqrt{5} - 1} = \dfrac{\sqrt{5} + 1}{4}
To find xx intercept
Put y=0y = 0 in (i)
⇒⇒ x=2+t2x = 2 + {t}^{2}
x=6x = 6
∵∵ Slope of common normal =−t=−2= - t = - 2
∴∴ Slope of tangent =12= \dfrac{1}{2}
Practice Parabola on Quantrex Academy →

More from Parabola

Let T be the tangent to the parabola y^2 = 16x at the point (64, 32) . Let L be the… JEE Advanced 2026Let O be the vertex of the parabola y^2=4x and its chords OP and OQ are perpendicular to… JEE Main 2026Let chord PQ of length 3 13 of the parabola y^2 = 12x be such that the ordinates of… JEE Main 2026Let the point P be the vertex of the parabola y = x^2 - 6x + 12 . If a line passing… JEE Main 2026Let the directrix of the parabola P: y^2 = 8x , cut x -axis at the point A . Let B(α, β)… JEE Main 2026Let A,B and C be the vertices of a variable right angled triangle inscribed in the… JEE Main 2026Consider the parabola P : y^2 = 4kx and the ellipse E : (x^2)/(a^2) + (y^2)/(b^2) = 1 .… JEE Main 2026Let the parabola y = x^2 + px + q passing through the point (1, -1) be such that the… JEE Main 2026Let A be the focus of the parabola y^ 2 =8 x . Let the line y=m x+c intersect the… JEE Main 2026Let the image of parabola x^ 2 =4 y , in the line x-y=1 be (y+a)^ 2 =b(x-c) , a, b, c ∈… JEE Main 2026An equilateral triangle OAB is inscribed in the parabola y^ 2 =4 x with the vertex O at… JEE Main 2026Let the locus of the mid-point of the chord through the origin O of the parabola y^ 2 =4… JEE Main 2026If the chord joining the points P_ 1 (x_ 1 , y_ 1 ) and P_ 2 (x_ 2 , y_ 2 ) on the… JEE Main 2026Let y^ 2 =12 x be the parabola with its vertex at O. Let P be a point on the parabola and… JEE Main 2026Let one end of a focal chord of the parabola y^ 2 =16 x be (16,16) . If P(α, β) divides… JEE Main 2026Let O be the vertex of the parabola x^ 2 =4 y and Q be any point on it. Let the locus of… JEE Main 2026 Full Parabola list Parabola JEE Maths PYQsJEE Mathematics PYQs All JEE Advanced PYQs