Question

Show that the equation of the common tangents to the circle $x^2+y^2=2a^2$ and the parabola $y^2=8ax$ are $y=\pm (x+2a)$

Answer 1

Given the equation of the parabola is $y^2=8ax$ ..............(1)
and the circle is $x^2+y^2=2a^2$................(2)
Any tangent of (1) is
$y=mx+\frac{2a}{m}\Rightarrow mx-y+\frac{2a}{m}=0$..........(3)
line (3) is also a tangent to circle (2)
$\Rightarrow \frac{|\frac{2a}{m}|}{\sqrt{(1+m^2)}}=a\sqrt{2}$
$\Rightarrow \frac{4}{m^2}=2(1+m^2)\Rightarrow m^4+m^2-2=0$
$\Rightarrow (m^2+2)(m^2-1)=0$ i.e
$\Rightarrow m^2\ne -2,m^2=1\Rightarrow m\pm 1$
The equation of the common tangent (1) & (2) are $y=\pm x\pm 2a\Rightarrow y=\pm (x+2a)$