Question

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

• A. $\frac{\sqrt{3}}{4}$
• B. $\frac{3\sqrt{2}}{8}$
• C. $\frac{8}{3\sqrt{2}}$
• D. $\frac{4}{\sqrt{3}}$

Answer 1

Answer: (B)

$\frac{3\sqrt{2}}{8}$

Let $P=(y^2,y)$ be a point on the curve.
The perpendicular distance from $P$ to $x-y+1=0$ is $\frac{|y^2-y+1|}{\sqrt{2}}$
Now, the discriminant for the expression $y^2-y+1$ is negative and the co-efficient of $y^2$ is positive. Hence, the expression $y^2-y+1$ is always positive.
For the minimum value of distance, $y=\frac{1}{2}$.
Hence,
Minimum distance $=\frac{\frac{3}{4}}{\sqrt{2}}=\frac{3\sqrt{2}}{8}$