The circle `x^2+y^2-8x = 0` and hyperbola `x^2 /9 - y^2 /4=1` intersect at the points A and B. Then the equation of the circle with AB as its diameter is

Options

A. `x^(2)+y^(2)-12x+24=0`
B. `x^(2)+y^(2)+12x+24=0`
C. `x^(2)+y^(2)+24x-12=0`
D. `x^(2)+y^(2)-24x-12=0`

Answer

A

Explanation

The coordinates of any point on the given hyperbola are `(3sectheta,2tantheta)`. If lies on the circle `x^(2)+y^(2)-8x=0`, then
`9sec^(2)theta+4tan^(2)theta-24sectheta=0`
`implies13sec^(2)theta-24sectheta-4=0`
`implies(13sectheta+2)(sectheta-2)=0`
`impliessectheta=2`, `-(2)/(13)impliessectheta=2`
`:.tantheta=+-sqrt(3)`
So, the coordinates of `A` and `B` are `(6,2sqrt(3))` and `(6,-2sqrt(3))`.
The equation of the circle with `AB` as its diameter is
`(x-6)^(2)+(y^(2)+12)=0` or, `x^(2)+y^(2)-12x+24=0`