Question

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

• A. $x{a}^{1/3}+y{b}^{1/3}+a^{2/3}{b}^{2/3}=0$
• B. $x{b}^{1/3}+y{a}^{1/3}+a^{2/3}{b}^{2/3}=0$
• C. $x{a}^{1/3}+y{b}^{1/3}-a^{2/3}{b}^{2/3}=0$
• D. None of these

Answer 1

Answer: (A)

Any tangent to the parabola $y^2=4ax$ is
$y=mx+\frac{a}{m}\dots (1)$
$\text{ meets }{x}^2=4by\text{ where }{x}^2=4b\left(mx+\frac{a}{m}\right)$
$\text{ or }{\left(mx\right)}^2-4{\left(bm\right)}^{2}x-4ab=0\dots (2)$
If the line ( $1$ ) be a tangent to the second parabola, then roots of ( $2$ ) must be equal. The condition for this is
$$\begin{gathered} 16b^2{m}^4+16abm=0\text{ or }{m}^3=-\frac{a}{b} \\ \therefore m=-{\left(\frac{a}{b}\right)}^{1/3}=-\frac{a^{1/3}}{b^{1/3}} \end{gathered}$$
Substituting this value of $m$ in $(1),$ we $get$
$$\begin{gathered} y=-\frac{a^{1/3}}{b^{1/3}}x+a\left(-\frac{b^{1/3}}{a^{1/3}}\right) \\ \text{ i. }e\cdot {\left(xa\right)}^{1/3}+{\left(yb\right)}^{1/3}+a^{2/3}{b}^{2/3}=0 \end{gathered}$$
which is the required equation of the common tangent.