Question

The equation of the common tangent to $x^2+y^2=2a^2$ and $y^2=8ax$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Equation of a tangent to $y^2=8ax$ is $y=mx+\frac{2a}{m}$
If this is a tangent to the circle $x^2+y^2=2a^2,then\left|\frac{\frac{2a}{m}}{\sqrt{1+m^2}}\right|=\sqrt{2}a\Rightarrow 2(1+m^2)=\frac{4}{m^2}$
$\Rightarrow m^4+m^2=2\Rightarrow m^4+m^2-2=0\Rightarrow (m^2{)}^2+2m^2-m^2-2=0\Rightarrow (m^2+2\left)\right(m^2-1)=0\Rightarrow m^2=1\Rightarrow m=\pm 1$
Common tangent is $y=\pm (x+2a)$.