Question

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

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

Answer 1

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

$y^2=4x$

$\Rightarrow a=1$ [Comparing with the standard equation $y^2=4ax$]

$x^2=-32y$

$\Rightarrow a=-8$ [Comparing with the standard equation $x^2=4ay$]

Tangent to the parabola $y^2=4x$ is $y=mx+\frac{a}{m}$

$\Rightarrow y=mx+\frac{1}{m}$----------------(1)

Tangent to the parabola $x^2=-32y$ is $y=mx+a{m}^2$

$\Rightarrow y=mx+8m^2$---------------(2)

Equating (1) & (2), we get

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

$\Rightarrow 8m^3=1$

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

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