If the two tangents drawn from a point P to the parabola y^2=4x are at right angles then find the locus of P.

daniko883y

daniko883y

Answered question

2022-09-29

If the two tangents drawn from a point P to the parabola y 2 = 4 x are at right angles then find the locus of P.

Answer & Explanation

Abigayle Lynn

Abigayle Lynn

Beginner2022-09-30Added 12 answers

The two points P 1 ( p 2 , 2 a p ) and P 2 ( 1 p 2 , 2 p ) lie on the parabola y 2 = 4 x.
Differentiating gives the slopes of tangents at P 1 , P 2 as 1 p , p respectively, i.e. perpendicular.
The equations of the tangents are:
(1) At  P 1 : p y = x + p 2 At  P 2 : y p = x + 2 p 2 (2) p y = p 2 x 1 ( 1 ) = ( 2 ) : ( 1 + p 2 ) x = ( 1 + p 2 ) x = 1
which is the equation of the locus of the point of intersection. This also happens to be the directrix. The fact that the locus of the point of intersection of two mutually perpendicular tangents is the directrix is a well-known property of the parabola.

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