Question

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

• A. $\frac{1}{2}$
• B. $\frac{1}{2\sqrt{2}}$
• C. $\frac{1}{\sqrt{2}}$
• D. 0

Answer 1

Answer: (B)

Shortest distance between curves is always along common normal.
${\frac{dy}{dx}|}_P=$ slope of line $=1$
$\Rightarrow x_0=1$
$\therefore y_0=\frac{1}{2}$
$\Rightarrow P\left(1,\frac{1}{2}\right)$
$\therefore$ Shortest distance
$=\left|\frac{1-\frac{1}{2}-1}{\sqrt{1^2+1^2}}\right|$
$=\frac{1}{2\sqrt{2}}$