Let P(asecθ,btanθ) and Q(asecϕ,btanϕ), where θ+ϕ=π2, be two points on the hyperbola x2a2−y2b2=1. If (h,k) is the point of intersection of normals at P and Q, then k is equal to
A
a2+b2a
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B
−(a2+b2a)
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C
a2+b2b
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D
−(a2+b2b)
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Solution
The correct option is D−(a2+b2b) Given P(asecθ,btanθ) and Q(asecϕ,btanϕ). The equation of normal at point P is a2xasecθ+b2ybtanθ=a2+b2 ⇒axsinθ+by=(a2+b2)tanθ⋯(1) Similarly the equation of normal at point Q is axsinϕ+by=(a2+b2)tanϕ⋯(2)
On mulitplying (1) by sinϕ and (2) by sinθ , we get axsinθsinϕ+bysinϕ=(a2+b2)tanθsinϕ axsinϕsinθ+bysinθ=(a2+b2)tanϕsinθ
On subtracting eqn (2) from (1), we get by(sinϕ−sinθ)=(a2+b2)(tanθsinϕ−tanϕsinθ) ∴y=k=a2+b2b⋅tanθsinϕ−tanϕsinθsinϕ−sinθ ∵θ+ϕ=π2⇒ϕ=π2−θ ⇒sinϕ=cosθandtanϕ=cotθ ∴y=k=a2+b2b⋅tanθcosθ−cotθsinθcosθ−sinθ =a2+b2b(sinθ−cosθcosθ−sinθ)=−(a2+b2)b