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 $\frac{1}{2}$

Parabolas $y^2=4x$ and $x^2=-32y$

$y^2=4x\Rightarrow$ $a=1$

$x^2=-32y$

$\Rightarrow$ $-48y\Rightarrow$ $A=8$

Here, lines are tangent

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

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

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

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

from (1) and (2)

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

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

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

Hence this is the answer