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 conjugate axis of the hyperbola is

Options

A. `4sqrt(11)`
B. `2sqrt(11)`
C. `4sqrt(22)`
D. `2sqrt(22)`

Answer

D

Explanation

Centre `-=(1,2)`
Radius of auxiliary circle `=a =sqrt((2-1)^(2)+(5-2)^(2))=sqrt(10)`
`2ae=sqrt(8^(2)+8^(2))=8sqrt2 or e=(4)/(sqrt5)`
`b^(2)=a^(2)e^(2)-a^(2)=32-10=22`
`"or "2b=2sqrt(22)`