Question

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

A
742
B
78
C
1142
D
2

Solution

## The correct option is A $$\dfrac{7}{4\sqrt{2}}$$we have ,$$2y.\dfrac{dy}{dx}=1\Rightarrow\dfrac{dy}{dx}]_{P(2+t^2,t)}=\dfrac{1}{2t}=1$$$$\Rightarrow t=\dfrac{1}{2}$$$$\therefore P\left(\dfrac{9}{4},\dfrac{1}{2}\right)$$So, shortest distance$$=\dfrac{\left|\dfrac{9}{4}-\dfrac{2}{4}\right|}{\sqrt{2}}=\dfrac{7}{4\sqrt{2}}$$Mathematics

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