Let $ P $ be a point on a variable hyperbola $ x^{2}-y^{2}=a^{2} $. Locus of $ P $ such that $ P $ is nearest to the line $ y=2 x $ is, a is a parameter

Question

Let $ P $ be a point on a variable hyperbola $ x^{2}-y^{2}=a^{2} $. Locus of $ P $ such that $ P $ is nearest to the line $ y=2 x $ is, a is a parameter, $ x+2 y=2 $, $ x=4 y $, $ x-2 y=0 $, $ 2 x+y=2 $

MARKS App · Accepted Answer

Let P h , k = ( a sec ⁡ θ , a tan ⁡ θ ) on x 2 - y 2 = a 2 2 x - 2 y d y d x = 0 ⇒ d y d x = x y , d x d y a t P = c o s e c θ For nearest point c o s e c θ = 2 , θ = π 6 ∴ h = a sec ⁡ π 6 , k = a tan ⁡ π 6 h = a 2 3 , k = a 1 3 , h = 2 k ⇒ x - 2 y = 0

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