Question

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

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

Answer 1

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

Let $PQ$ be the shortest distance between the line $y=x$ and the curve $y^2=x-2$.

Slope of the line $y=x$ is $1$.
$\frac{dy}{dx}$ at point $P$ will be $1$ as both are parallel.
Differentiating the curve $y^2=x-2$ w.r.t. $x$, we get
$2y{y}^{\prime }=1$
$\Rightarrow y^{\prime }=\frac{1}{2y}=\frac{1}{2\alpha }=1$
$\therefore P=(\frac{9}{4},\frac{1}{2})$
Minimum diatance is given by,
$PQ=\left|\frac{\frac{9}{4}-\frac{1}{2}}{\sqrt{2}}\right|$
$=\frac{7}{4\sqrt{2}}$