Equation of a circle in a parabola

Geometry Level 3

A circle touches the parabola y 2 = 4 a x y^{2} = 4ax at a point P P . It also passes through the focus F F of the parabola and intersects its horizontal axis at Q Q . If ∠ F P Q \angle FPQ is 9 0 ∘ 90^\circ , and the equation of the circle can be written as ( x − k a ) 2 + y 2 = ( 4 a ) 2 (x - ka)^{2} + y^{2} = (4a)^{2} then find k k .


The answer is 5.

Vijay Simha by Vijay Simha , 47, USA

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1 solution

Rab Gani
Mar 29, 2018

Ans. 5

The circle has a center at C(ka,0), and radius r=4a. Since the point C, F and Q are on the x-axis and the line drawn through them are the diameter of the circle, (<FPQ = 90 ), then we can find the the radius r = (ka – a) = 4a, then k =5

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