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At the point of intersection of the rect...

At the point of intersection of the rectangular hyperbola `xy=c^2` and the parabola `y^2=4ax` tangents to the rectangular hyperbola and the parabola make angles `theta` and `phi` , respectively with x-axis, then

A

`theta = tan^(-1) (-2 tan phi)`

B

`theta = (1)/(2) tan^(-1)(-tan phi)`

C

`phi = tan^(-1)(-2tan theta)`

D

`phi =(1)/(2) tan^(-1)(-tan theta)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `(x_(1),y_(1))` be point of intersection
`rArr y_(1)^(2) = 4ax_(1), x_(1)y_(1) = c^(2)`
`y^(2) = 4ax xy = c^(2)`
`m_(T(x_(1),y_(1))) =(2a)/(y_(1)) = tan phi`
`M_(T(x_(1),y_(1))) =-(y_(1))/(x_(1)) =tan theta`
So `(tan theta)/(tan phi) = (-y_(1)^(2))/(2ax_(1)) =- (4ax_(1))/(2ax_(1)) =-2`
So `theta = tan^(-1) (-2 tan phi)`
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