Question

Two common tangents to the circle $x^2+y^2=2a^2$ and parabola $y^2=8ax$ are

• A. $x=\pm (y+2a)$
• B. $y=\pm (x+2a)$
• C. $x=\pm (y+a)$
• D. $y=\pm (x+a)$

Answer 1

Answer: (B)

The equation of any tangent to $y^2=8ax$ is
$y=mx+\frac{2a}{m}$ ... (i)
If it touches $x^2+y^2=2a^2$, then
${\left(\frac{2a}{m}\right)}^2=2a^2\left(1+m^2\right)$
$[\because c^2=a^2(1+m^2)]$
$\Rightarrow 2=m^2\left(m^2+1\right)$
$\Rightarrow m^4+m^2-2=0$
$\Rightarrow \left(m^2+2\right)\left(m^2-1\right)\equiv 0$
$\Rightarrow m^2-1=0$
$\Rightarrow m=\pm 1$
Putting the values of $m$ in Eq. (i), we get $y=\pm (x+2a)$ as the equations of common tangents.