Question

Equation as 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

Answer: (B)

$2x-\sqrt{5}y+4=0$

Let tangent to $\frac{x^2}{9}-\frac{y^2}{4}=1$
is $y=mx+\sqrt{{9m}^2-4}$ $m>0$
It is tangent to $x^2+y^2-8x=0$
$\therefore \frac{4m+\sqrt{{9m}^2-4}}{\sqrt{1+m^2}}=4\Rightarrow 495m^4+104m^2-400=0$
$\Rightarrow m^2=\frac{4}{5}\Rightarrow m=\frac{2}{\sqrt{5}}$
Therefore the tangent is
$y=\frac{2}{\sqrt{5}}m+\frac{4}{\sqrt{5}}\Rightarrow 2x-\sqrt{5y}+4=0$