Question

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

• A. $\frac{2\sqrt{3}}{8}$
• B. $\frac{3\sqrt{2}}{5}$
• C. $\frac{\sqrt{3}}{4}$
• D. $\frac{3\sqrt{2}}{8}$

Answer 1

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

Let $(a^2,a)$ be the point of shortest distance on $x=y^2$. Then distance between $(a^2,a)$ and line $x-y+1=0$ is given by
$D=\frac{a^2-a+1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\left[\right(a-\frac{1}{2}{)}^2+\frac{3}{4}]$
It is min when $a=\frac{1}{2}\text{and}{D}_{min}=\frac{3}{4\sqrt{2}}=\frac{3\sqrt{2}}{8}$