Question

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

• A. $\frac{3\sqrt{2}}{8}$
• B. $\frac{2\sqrt{3}}{8}$
• C. $\frac{3\sqrt{2}}{5}$
• D. $\frac{\sqrt{3}}{4}$

Answer 1

Answer: (A)

$x-y+1=0\dots \left(1\right)$
$x=y^2$
$1-2y\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=\frac{1}{2y}=$ Slope of given line $\left(1\right)$
$\frac{1}{2y}=1\Rightarrow y=\frac{1}{2}\Rightarrow x={\left(\frac{1}{2}\right)}^2=\frac{1}{4}\Rightarrow \left(x,y\right)=\cdot \left(\frac{1}{4},\frac{1}{2}\right)$
$\therefore$ The shortest distance is $\frac{|\frac{1}{4}-\frac{1}{2}+1|}{\sqrt{1+1}}=\frac{3}{4\sqrt{2}}=\frac{3\sqrt{2}}{8}$
Directions: Question number 86 to 90 are Assertion - Reason type questions. Each of these questions contains two statements
Statement-1 (Assertion) and Statement-2 (Reason).
Each of these questions also have four alternative choices, only one of which is the correct answer. You have to select the correct choice