Question

The equation of the common tangent to the parabolas $y^{2}=2x$ and $x^{2}=16y$ is

• A. $2x+y-2=0$
• B. $x-2y-2=0$
• C. $x-2y+2=0$
• D. $x+2y+2=0$

Answer 1

Answer: (D)

Any tangent to $y^{2}=2x$ in slope form is
$y=mx+\frac{1}{2 m}$ which is also tangent to $x^{2}=16y$
$\Rightarrow x^{2}=16\left(m x + \frac{1}{2 m}\right)$ will have equal roots.
$\Rightarrow $ Discriminant of $x^{2}-16mx-\frac{8}{m}=0$ is zero
$\Rightarrow \left(16 m\right)^{2}-4\times 1\left(- \frac{8}{m}\right)=0$
$\Rightarrow m^{3}=-\frac{1}{8}\Rightarrow m=-\frac{1}{2}$
$\Rightarrow $ Equation of the common tangent is $y=-\frac{x}{2}-1$
$\Rightarrow x+2y+2=0$