Question

The slope of the line touching both the 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 $\frac{1}{2}$

Equation of tangent at $A(t^2,2t)$

$yt=x+t^2$ is tangent to $x^2+32y=0$ at $B$

$\Rightarrow x^2+32(\frac{x}{t}+t)=0$

$\Rightarrow x^2+\frac{32}{t}x+32t=0$

$\Rightarrow (\frac{32}{t}{)}^2-4(32t)=0$

$\Rightarrow 32(\frac{32}{t^2}-4t)=0$

$\Rightarrow t^3=8\Rightarrow t=2$

$\Rightarrow$ Slope of tangent is $\frac{1}{t}$=$\frac{1}{2}$