Question

The shortest distance between the line $x-y=1$ and the curve $x^{2}=2 y$ 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.
$\left.\frac{ dy }{ dx }\right|_{ P }=$ 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
$=\left|\frac{1-\frac{1}{2}-1}{\sqrt{1^{2}+1^{2}}}\right|$
$=\frac{1}{2 \sqrt{2}}$