Question

Show that the common tangent to the parabola $y^2=4ax$ and $x^2=4by$ is $x{a}^{1/3}+y{b}^{1/3}+a^{2/3}{b}^{2/3}=0$

Answer 1

Equation of tangent to $y^2=4ax$ :

$y=mx+\frac{a}{m}---\left(1\right)$

Equation of tangent to $x^2=4by$:

$x=ky+\frac{b}{K}$

$\Rightarrow y=\frac{x}{K}-\frac{b}{K^2}----\left(2\right)$

For Common tangent $\left(1\right)$ & $\left(2\right)$ must represent same equation

$\Rightarrow m=\frac{1}{K};\frac{a}{m}=\frac{-b}{K^2}$

$\Rightarrow aK=\frac{-b}{K^2}$

$\Rightarrow K^3=\frac{-b}{a}$

$\Rightarrow K=\frac{-b^{1/3}}{a^{1/3}}$

Equation of common tangent:

$K^{2}y-Kx+b=0$

$\Rightarrow \frac{b^{2/3}}{a^{2/3}}y+\frac{b^{1/3}}{a^{1/3}}x+b=0$

$\Rightarrow x{a}^{1/3}+y{b}^{1/3}+a^{2/3}{b}^{2/3}=0$