A pair of tangents with inclinations α , β are drawn from an external point P to the parabola y 2 = 16 x . If the point P varies in such a way that tan 2 α + tan 2 β = 4 then the locus of P is a conic

Question

A pair of tangents with inclinations α , β are drawn from an external point P to the parabola y 2 = 16 x . If the point P varies in such a way that tan 2 α + tan 2 β = 4 then the locus of P is a conic whose eccentricity is

TestHub · Accepted Answer

Let m 1 ​ = tan α , m 2 ​ = tan β , Let P = ( h , k ) m 1 ​ , m 2 ​ are the roots of K = mh + m 4 ​ ⇒ h m 2 − K m + 4 = 0 m 1 ​ + m 2 ​ = h K ​ ; m 1 ​ m 2 ​ = h 4 ​ m 1 2 ​ + m 2 2 ​ = h 2 K 2 ​ − h 8 ​ = 4 Locus of P is y 2 − 8 x = 4 x 2 ⇒ y 2 = 4 ( x + 1 ) 2 − 4 ⇒ 1 ( x + 1 ) 2 ​ − 4 y 2 ​ = 1

Mathematics - Parabola Question with Solution | TestHub

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