#### Question

Find the coordinates of a point on the parabola y=x^{2}+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?

#### Solution

\[\text { Let coordinates of the point on the parabola be } \left( x, y \right) . \text { Then }, \]

\[y = x^2 + 7x + 2 ............. \left( 1 \right)\]

\[\text { Let the distance of a point } \left( x, \left( x^2 + 7x + 2 \right) \right) \text { from the line } y = 3x - 3\text { be S . Then,} \]

\[S = \left| \frac{- 3x + \left( x^2 + 7x + 2 \right) + 3}{\sqrt{10}} \right|\]

\[ \Rightarrow \frac{dS}{dt} = \frac{- 3 + 2x + 7}{\sqrt{10}}\]

\[\text { For maximum or minimum values of S, we must have }\]

\[\frac{dS}{dt} = 0\]

\[ \Rightarrow \frac{- 3 + 2x + 7}{\sqrt{10}} = 0\]

\[ \Rightarrow 2x = - 4\]

\[ \Rightarrow x = - 2\]

\[\text { Now }, \]

\[\frac{d^2 S}{d t^2} = \frac{2}{\sqrt{10}} > 0\]

\[\text { So, the nearest point is} \left( x, \left( x^2 + 7x + 2 \right) \right) . \]

\[ \Rightarrow \left( - 2, 4 - 14 + 2 \right)\]

\[ \Rightarrow \left( - 2, - 8 \right)\]