Question

Find the equations to the common tangents of the parabolas $y^2=4axand{x}^2=4by$

Answer 1

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

$y=mx+\frac{a}{m}...............\left(i\right)$

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

$x=m_{1}y+\frac{b}{m_1}\Rightarrow y=\frac{1}{m_1}x-\frac{b}{(m_1{)}^2}............\left(ii\right)$

for common tangent (i) & (ii) must represent the same line

$\frac{1}{m_1}=m\&\frac{a}{m}=-\frac{b}{m_1^2}$

$\Rightarrow \frac{a}{m}=-b{m}^2\Rightarrow m={(-\frac{a}{b})}^{\frac{1}{3}}$

$\therefore$ equation of common tangent is

$y={(-\frac{a}{b})}^{\frac{1}{3}}x+a{(-\frac{b}{a})}^{\frac{1}{3}}$