Question

The slope of the line touching both the parabolas $y^2=4x$ and $x^2=-32y$ is:

• A. $\frac{1}{8}$
• B. $\frac{2}{3}$
• C. $\frac{1}{2}$
• D. $\frac{3}{2}$

Answer 1

The correct option is C $\frac{1}{2}$

$y^2=4x$

$\Rightarrow a=1$

$x^2=-32y$

$\Rightarrow a=-8$

Tangent to the parabola $y^2=4x$ is

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

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

tangent to the parabola $x^2=-32y$ is

$y=mx-a{m}^2$

$y=mx+8m^2$

$\therefore mx+\frac{1}{m}=mx+8m^2$

$8m{x}^3=1$

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

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