Question

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

• A. $\frac{7}{4\sqrt{2}}$
• B. $\frac{7}{8}$
• C. $\frac{11}{4\sqrt{2}}$
• D. $2$

Answer 1

The correct option is A $\frac{7}{4\sqrt{2}}$

we have ,$2y.\frac{dy}{dx}=1\Rightarrow \frac{dy}{dx}{]}_{P(2+t^2,t)}=\frac{1}{2t}=1$
$\Rightarrow t=\frac{1}{2}$
$\therefore P(\frac{9}{4},\frac{1}{2})$
So, shortest distance
$=\frac{|\frac{9}{4}-\frac{2}{4}|}{\sqrt{2}}=\frac{7}{4\sqrt{2}}$