Question

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

• A. $x+2y+2=0$
• B. $x+2y+1=0$
• C. $2x+y+1=0$
• D. $Noneofthese$

Answer 1

The correct option is A $x+2y+2=0$

equation of tangent to the parabola

$y^2=2x$ is $y=mx+\frac{\frac{1}{2}}{m}$

$\Rightarrow y=mx+\frac{1}{2m}$ _______ (i)

And equation of tangent to the parabola

$x^2=16y$ is $y=mx-4m^2$ _________ (ii)

For common tangent, equation (i) and equation (ii) are identical.

$\therefore \frac{1}{2m}=-4m^2$

$\Rightarrow m^3=-\frac{1}{8}$

$\Rightarrow m=-\frac{1}{2}$

$\therefore$ equation of common tangent is

$y=-\frac{1}{2}x-\frac{\frac{1}{2}}{\frac{1}{2}}$

$\Rightarrow 2y=-x-2$

$\Rightarrow x+2y+2=0$