Question

The common tangent to the parabola $y^2=4ax$ and $x^2=4ay$ is

• A. $x+y+a=0$
• B. $x+y-a=0$
• C. $x-y+a=0$
• D. $x-y-a=0$

Answer 1

The correct option is A $x+y+a=0$

The equation of any tangent to $y^2=4ax$ is $y=mx+\frac{a}{m}$, if it touches $x^2=4ay$,

then the equation $x^2=4a(mx+\frac{a}{m})$ has equal roots or, $m{x}^2-4a{m}^{2}x-4a^2=0$ has equal roots

$\Rightarrow 16a^2{m}^4=-16a^{2}m\Rightarrow m=-1(\because m\ne 0)$

Putting $m=-1$ in $y=mx+\frac{a}{m}$, we get $y=-x-a$ or, $x+y+a=0$