Question

The equation to the line touching both the parabolas $y^2=4x$ and $x^2=-32y$, is

• A. $x+2y+4=0$
• B. $2x+y-4=0$
• C. $x-2y-4=0$
• D. $x-2y+4=0$

Answer 1

The correct option is D $x-2y+4=0$

Equation of tangent

$y^2=4x$

$\Rightarrow y=mx+\frac{1}{m}$ ......$\left(1\right)$

Equation of tangent

$x^2=-32y$

$y=mx-(-8m{)}^2$

$a=-8$

$y=mx-(-8m{)}^2$

$y=mx+8m^2$ ......$\left(2\right)$

from equation $\left(1\right)$ & $\left(2\right)$

$\frac{1}{m}=8m^2$

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

$m^3={\left(\frac{1}{2}\right)}^3$

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

$y=mx+\frac{1}{m}$

equation of tangent $y=\frac{x}{2}+2$

$y=\frac{x+4}{2}$

$\Rightarrow 2y=x+4$

$\Rightarrow x-2y+4=0$

$\therefore$ The given option is correct.