The circle `x^(2)+y^(2)-8x=0` and hyperbola `(x^(2))/(9)-(y^(2))/(4)=1` intersect at the points `A` and `B`. The equation of a common tangent with positive slope to the circle as well as to the hyperbola, is

Options

A. `2x-sqrt(5)y-20=0`
B. `2x-sqrt(5)y+4=0`
C. `3x-4y+8=0`
D. `4x-3y+4=0`

Explanation

The equation of a tangent of slope `m` to the hyperbola `(x^(2))-(9)-(y^(2))/(4)=1` is
`y=mx+sqrt(9m^(2)-4)`
If it touches the circle `x^(2)+y^(2)-8x=0`, then
`|(4m+sqrt(9m^(2)-4))/(sqrt(1+m^(2)))|=4`
`implies495m^(4)+104m^(2)-400=0`
`implies(5m^(2)-4)(99m^(2)+100)=0`
`impliesm^(2)=(4)/(5)=(2)/(sqrt(5))`
Substituting the value of `m` in `(i)`, we get
`2x-sqrt(5)y+4=0`
as the equation of the required common tangent