Single Choice

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

A$$\dfrac{7}{4\sqrt{2}}$$
Correct Answer
B$$\dfrac{7}{8}$$
C$$\dfrac{11}{4\sqrt{2}}$$
D$$2$$

Solution

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
$$=\dfrac{\left|\dfrac{9}{4}-\dfrac{2}{4}\right|}{\sqrt{2}}=\dfrac{7}{4\sqrt{2}}$$

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