Question

Find the equation of the common tangent to the parabolas $y^2=4ax$ and $x^2=4by.$

• A. $y{b}^{1/3}+x{a}^{1/3}+a^{2/3}{b}^{2/3}=0.$
• B. $y{b}^{1/4}+x{a}^{1/3}+a^{1/3}{b}^{1/3}=0.$
• C. $y{b}^{1/3}+x{a}^{1/4}+a^{2/3}{b}^{1/3}=0.$
• D. $y{b}^{1/3}+x{a}^{1/3}+a^{1/3}{b}^{2/5}=0.$

Answer 1

The correct option is A $y{b}^{1/3}+x{a}^{1/3}+a^{2/3}{b}^{2/3}=0.$

Any tangent to the parabola $y^2=4ax$ is
$y=mx+\frac{a}{m}...\left(1\right)$
If it is a tangent to the parabola $x^2=4by,$ then it will cut it in two coincident points. Eliminating y, we get
$x^2=4b(mx+\frac{a}{m})$
or $x^2-4bmx-4b\frac{a}{m}=0...\left(2\right)$
The roots of (2) will be equal if $B^2-4AC=0$
or $16b^2{m}^2+\frac{16ab}{m}=0$
or $m^3=-\frac{a}{b}\therefore m=-\frac{a^{1/3}}{b^{1/3}}$
On putting the value of $m$ in $\left(1\right)$, the equation of the common tangent is
$y=-\frac{a^{1/3}}{b^{1/3}}x-a.\frac{b^{1/3}}{a^{1/3}}$
or $y{b}^{1/3}+x{a}^{1/3}+a^{2/3}{b}^{2/3}=0.$