Question

The shortest distance between the line y - x = 1and the curve x = y² is

विकल्प

• \[\frac{3\sqrt{2}}{8}\]

• \[\frac{2\sqrt{3}}{8}\]

• \[\frac{3\sqrt{2}}{5}\]

• \[\frac{\sqrt{3}}{4}\]

Answer 1

\[\frac{3\sqrt{2}}{8}\]

Explanation:

Given the equation of the line is 

x − y + 1 = 0...(i)

∴ Slope = 1
x = y²

Differentiating w.r.t. x, we get

\[1=2y\frac{\mathrm{d}y}{\mathrm{d}x}\]

\[\therefore\quad\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{2y}\]

Is the same as the slope of (i)

\[\therefore\quad\frac{1}{2y}=1\]

\[\therefore\quad y=\frac{1}{2}\]

\[\therefore\quad\mathrm{P}(x,y)=\left(\frac{1}{4},\frac{1}{2}\right)\]

∴ Shortest distance = \[\frac{\left|\frac{1}{4}-\frac{1}{2}+1\right|}{\sqrt{1+1}}\] \[=\frac{3\sqrt{2}}{8}\]