At the point of intersection of the rectangular hyperbola xy=c2 and the parabola y2=4ax tangents to the rectangular hyperbola and the parabola make an angle θ and ϕ respectively with the axis of X , then
A
θ=tan−1(−tanϕ)
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B
ϕ=tan−1(−tanθ)
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C
θ=12tan−1(−tanϕ)
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D
ϕ=12tan−1(−tanθ)
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Solution
The correct option is Aθ=tan−1(−tanϕ)
Given hyperbola xy=c2 parabola y2=4ax
point of intersection of hyperbola and parabola
⇒y2=4a(c2y) ⇒y3=4ac2 ⇒y=3√4ac2 ⇒x=c23√4ac2
Point P : (c23√4ac2,3√4ac2)
Equation of tangent at P on hyperbola, x3√4ac2+y(c23√4ac2)2=c2