The locus of the foot of perpendicular from my focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consider the foci of a hyperbola as `(-3, -2)` and (5,6) and the foot of perpendicular from the focus (5, 6) upon a tangent to the hyperbola as (2, 5).
The directrix of the hyperbola corresponding to the focus (5, 6) is

Options

A. `(2//9,31//3)`
B. `(7//4,23//4)`
C. `(2//3,9)`
D. `(7//9,7)`

Answer

C

Explanation

The tangent is
`y-5=-(5-2)/(6-5)(x-2)`
`"or "3x+y=11`
The hyperbola is
`(x-5)^(2)+(y-6)^(2)=(16)/(5)xx((2x+2y-11)^(2))/(8)`
Solving, we get
`(x-5)^(2)+(5-3x)^(2)=(2)/(5)(2dx+22-6x-11)^(2)`
`"or "5[10x^(2)-40x+50]=2(11-4x)^(2)`
`"or "9x^(2)-12x+4=0`
`"or "(3x-2)^(2)=0`
`"or "x=(2)/(3),y=9`