The shortest distance between the

Question:

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

  1. (1) 2

  2. (2) $\frac{7}{8}$

  3. (3) $\frac{7}{4 \sqrt{2}}$

  4. (4) $\frac{11}{4 \sqrt{2}}$


Correct Option: , 3

Solution:

The shortest distance between line $y=x$ and parabola $=$ the distance $L M$ between line $y=x$ and tangent of parabola having slope 1 .

$y=m(x-2)+\frac{a}{m}$

Here $m=1$ and $a=\frac{1}{4}$

$\therefore$ equation of tangent is: $y=x-\frac{7}{4}$

Distance between the line $y-x=0$ and $y-x+\frac{7}{4}=0$

$=\left|\frac{\frac{7}{4}-0}{\sqrt{1^{2}+1^{2}}}\right|=\frac{7}{4 \sqrt{2}}$

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