If the angle between the asymptotes of hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` is `120^0` and the product of perpendiculars drawn from the foci upon its any tangent is 9, then the locus of the point of intersection of perpendicular tangents of the hyperbola can be `x^2+y^2=6` (b) `x^2+y^2=9` `x^2+y^2=3` (d) `x^2+y^2=18`

Options

A. `x^(2)+y^(2)=6`
B. `x^(2)+y^(2)=9`
C. `x^(2)+y^(2)=3`
D. `x^(2)+y^(2)=18`

Answer

D

Explanation

The product of perpendiculars drawn from the foci upon any of its tangents is 9. Therefore,
`b^(2)=9`
Also, `(b)/(a)=tan30^(@)=(1)/(sqrt3)`

`ttherefore" "a^(2)=3b^(2)=27`
Therefore, the required locus is the director circle of the hyperbola which is given by
`x^(2)+y^(2)=27-9`
`"or "x^(2)+y^(2)=18`
If `b//a=tan 60^(@)`, then
`a^(2)=(b^(2))/(3)=(9)/(3)=3`
Hence, the required locus is `x^(2)+y^(2)=3-9=-6` which is not possible.