Question

Find the shortest distance between line $y=x-2$ and $y=x^2+3x+2$

Answer 1

Given ,

$y=x-2$

$x-y-2=0$ .....(1)

$y=x^2+3x+2$ ....(2)

Let $(t^2,t)$ be a point on $y=x^2+3x-2$ its distance to the line $y=x-2$ or$x-y-2=0$

$d=\left|\frac{t^2-t-2}{\sqrt{a^2+b^2}}\right|$

$d=\left|\frac{t^2-t-2}{\sqrt{{\left(1\right)}^2+{(-1)}^2}}\right|$

$d=\left|\frac{t^2-t-2}{\sqrt{2}}\right|$

Hence,

Required distance is $=\left|\frac{-2}{\sqrt{2}}\right|=\left|\frac{2}{\sqrt{2}}\right|=\left|\sqrt{2}\right|=\sqrt{2}$