Question

The shortest distance between the parabolas $y^2=4x$ and $y^2=2x-6$ is

• A. $2$
• B. $\sqrt{5}$
• C. $3$
• D. none of these

Answer 1

Answer: (B)

Normal to $y^2=4x$ at $\left(m^2,2m\right)$ is
$y+mx-2m-m^3=0$
Normal to $y^2=2(x-3)$ at $\left(1/2t^2+3,t\right)$ is
$y+t(x-3)-t-\frac{1}{2}{t}^3=0$
Both are same if $-2m-m^3=-4m-1/2m^3$
$\Rightarrow m=0,\pm 2$
So, points will be $(4,4)$ and $(5,2)$
Hence, shortest distance will be $=\sqrt{1+4}=\sqrt{5}$.