Question

The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at the points A and B.
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

• A. $2x-5\sqrt{y}-20=0$
• 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
$y=mx+\sqrt{9m^2-4}$ ...(i)
Equation of tangent to circle is
$y=m(x-4)+\sqrt{16m^2+16}$ ...(ii)
Eqs. (i) and (ii) will be identical for $m=\frac{2}{\sqrt{5}}$ satisfy.
$\therefore$ Equation of common tangent is $2x-\sqrt{5}y+4=0$.