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Question

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=tan1(tanϕ)
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B
ϕ=tan1(tanθ)
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C
θ=12tan1(tanϕ)
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D
ϕ=12tan1(tanθ)
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Solution

The correct option is A θ=tan1(tanϕ)
Given hyperbola xy=c2 parabola y2=4ax

point of intersection of hyperbola and parabola

y2=4a(c2y)
y3=4ac2
y=34ac2
x=c234ac2

Point P : (c234ac2,34ac2)

Equation of tangent at P on hyperbola, x34ac2+y(c234ac2)2=c2

4ac2x+c2y=2c2y=4ax+2

Slope = tanθ =4a----------(1)

equation of tangent at P on parabola ,
y(34ac2)=4a[(c234ac2)+x]y=4a(c24ac2+x34ac2)y=4ax+4ac24ac2

Slope =4a = tanϕ----------------(2)

From equation (1) and (2),we get,

tanθ=tanϕ

θ=tan1(tanϕ)

996514_1044552_ans_374cd4fb568b45f09a1435771457b744.png

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