question_answer

Shortest distance between the curve \[{{y}^{2}}=x-2\] and \[y=x\] is:

A) greater than 4

B) less than 2

C) greater than 3

D) greater than 2

Correct Answer: B

Solution :

Shortest distance between

\[{{y}^{2}}=x-2\]and \[y=x\]

\[\frac{dy}{dx}\]at point P will be 1

Differentiating the curve

\[2yy'=1\]\[\Rightarrow \]\[y'=\frac{1}{2y}=\frac{1}{2\alpha }=1\]

\[\therefore \]      \[P\left( \frac{9}{4},\frac{1}{2} \right)\]

minimum distance\[=PQ\]

\[=\left| \frac{\frac{9}{4}-\frac{1}{2}}{\sqrt{2}} \right|=\frac{7}{4\sqrt{2}}.\]