Question

Let P be the point on the parabola y (square) = 4x which is at the shortest distance from the centre S of the circle x (square) + y (square ) - 4x - 16x + 64 = 0. Let Q be the point on the circle dividing the line segement SP internally.

Answer 1

Hi,

Please refer the attached file !

Answer 2

For minimum distance from the centre of circle to the parabola at point P, the line must be normal to the parabola at P.

Equation of the parabola: y2 = 4ax = 8x

a = 2

Let P(at2, 2at) = (2t2, 4t)

Equation of normal to parabola is

y = –tx + 2at + at3

y = –tx + 4t + 2t3

It passes through centre of circle C(0, –6)

–6 = 4t + 2t3

t3 + 2t + 3 = 0

t = –1

Hence P is (2, –4), which is centre of required circle.

Radius of required circle = Distance between C and P.

r2 = (2-0)2 + (-4+6)2 = 4+4 = 8

Equation of required circle:

(x – 2)2 + (y + 4)2 = 8

x2 + y2 – 4x + 8y + 12 = 0