Question

The circle $x^2+y^2-8x=0$ and hyperbola$\frac{x^2}{9}-\frac{y^2}{4}=1$ intersects 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 :

• A. $2x-\sqrt{5}y-20=0$
• B. $2x-\sqrt{5}y+4=0$
• C. $3x-4y+8=0$
• D. $4x-3y+4=0$

Answer 1

The correct option is B $2x-\sqrt{5}y+4=0$

Equation of tangent to hyperbola having slope $m$ is $y=mx\pm \sqrt{9m^2-4}$ ...(1)
Equation of tangent to circle is $y=m(x-4)\pm \sqrt{16m^2+16}$ ...(2)
Equation (1) and (2) will be identical for $m=\frac{2}{\sqrt{5}}$
Therefore, equation of common tangent is $2x-\sqrt{5}y+4=0$