The Coordinates Of The Point On The Parabola y2 = 8x which Is At Minimum Distance From The Circle x2 + (y + 6)2 = 1 Are

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Question

The coordinates of the point on the parabola y² = 8x which is at minimum distance from the circle x² + (y + 6)² = 1 are

medium

Solution

Correct option is

(2, –4)

Let P (2t², 4t) be any point on the parabola. The centre of the given circle is O (0, –6) and the radius is 1.

(other roots are imaginary)

Hence OP² is minimum at t = –1. But if A is any point on the circle and onOP (min), then AP will be minimum when OP is minimum as AP = OP – (radius of circle). Thus the required point is P(2(–1))², 4(–1) = (2, –4).