Question

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

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

Answer 1

Answer: (B)

The equation of any tangent to $y^{2}=8 a x$ is
$y=m x+\frac{2 a}{m}$ ... (i)
If it touches $x^{2}+y^{2}=2 a^{2}$, then
$\left(\frac{2 a}{m}\right)^{2} =2 a^{2}\left(1+m^{2}\right)$
$[\because c^2 = a^2 (1 + m^2)]$
$\Rightarrow 2 =m^{2}\left(m^{2}+1\right)$
$\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 $y=\pm(x+2 a)$ as the equations of common tangents.