Find the equation of the common tangents Find the equation of the common tangents to the parabola $y^2=4ax$ and $x^2=4by$. 
My Attempt :
$y^2=4ax$ is the equation of parabola with focus at $(a,0)$ 
$x^2=4by$ is the equation of parabola with focus at $(0,b)$
The equation of the tangent to the parabola $y^2=4ax$ is 
$$y=mx+\dfrac {a}{m}$$
$$m^2x-my+a=0$$
The equation of the tangent to the parabola $x^2=4by$ is:
$$x=m'y+\dfrac {b}{m'}$$
$$(m')^{2}y - m'x+b=0$$

Let $A(at^2,2at)$ and $B(2bu,bu^2)$ be the points on $C_1:y^2=4ax$ and $C_2:x^2=4by$ respectively.
Equation of tangent of $C_1$ at $A$:
$$\frac{x}{-at^2}+\frac{y}{at}=1 \quad \cdots \cdots \: (1)$$
Equation of tangent of $C_2$ at $B$:
$$\frac{x}{bu}+\frac{y}{-bu^2}=1 \quad \cdots \cdots \: (2)$$
Comparing $(1)$ and $(2)$,
$$
\left \{
  \begin{array}{rcl}
    -at^2 &=& bu \\
    at &=& -bu^2 \\
  \end{array}
\right.$$
On solving,
$$t=\frac{1}{u}=-\sqrt[3]{\frac{b}{a}}$$
The common tangent is
$$-\frac{x}{\sqrt[3]{ab^2}}-\frac{y}{\sqrt[3]{a^2b}}=1$$
$$\fbox{$\sqrt[3]{a}\, x+\sqrt[3]{b} \, y+\sqrt[3]{a^2 b^2}=0$}$$
