Question

The shortest distance between line $y-x=1$ and curve $x=y^2$ is

• A. $\frac{3\sqrt{2}}{8}$
• B. $\frac{8}{3\sqrt{2}}$
• C. $\frac{4}{\sqrt{3}}$
• D. $\frac{\sqrt{3}}{4}$

Answer 1

The correct option is A $\frac{3\sqrt{2}}{8}$

Shortest distance between two curve occurred along the common normal Slope of normal to $y^2=x$ at point P$(t^2,t)$ is -2t and slope of line $y-x=1$ is 1.
As they are perpendicular to each other
$\therefore (-2t)=-1\Rightarrow t=\frac{1}{2}\therefore (\frac{1}{4},\frac{1}{2})$
And shortest distance $=\left|\frac{\frac{1}{2}-\frac{1}{4}-1}{\sqrt{2}}\right|$
So shortest distance between them is$\frac{3\sqrt{2}}{8}$