Question

Find equation of common tangent of $y^2=4ax$ and $x^2=4ay$.

Answer 1

equation of tangent to $y^2=4ax$

is $y=mx+\frac{a}{m}$

If this touches $x^2=4ay$ then,

$x^2=4a(mx+\frac{a}{m})$ has equal roots

$\Rightarrow m{x}^2-4a{m}^{2}x-4a^2=0$

as it has equal roots

$\Rightarrow 16a^2{m}^4=-16a^{2}m$

$\therefore m=-1$

putting m=-1 in $y=mx+\frac{a}{m}$

$\Rightarrow y=-x-a$ or

x+y+a=0