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

Answer: (A)

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