Question

The equation of the common tangent to the parabolas $y^2=4ax$ and $x^2=4by$ is-

• A. $x{a}^{1/3}+y{b}^{1/3}+{\left(ab\right)}^{2/3}=0$
• B. $x{a}^{2/3}+y{b}^{2/3}+{\left(ab\right)}^{2/3}=0$
• C. $x{a}^{1/3}+y{b}^{1/3}={\left(ab\right)}^{2/3}$
• D. $x{a}^{2/3}+y{b}^{2/3}={\left(ab\right)}^{2/3}$

Answer 1

The correct option is A $x{a}^{1/3}+y{b}^{1/3}+{\left(ab\right)}^{2/3}=0$

equation of tangent to $y^2=4ax$ is $y=mx+\frac{a}{m}$ ________ (1)

And equation of tangent to $x^2=4by$ is $y=mx-b{m}^2$ _________ (2)

For common tangents to the parabolas.

equation (i) and (ii) are identical.

$\therefore \frac{a}{m}=-b{m}^2$

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

$\therefore$ equation of common tangent is

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

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