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