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Question
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Solution
\[\text { Let coordinates of the point on the parabola be } \left( x, y \right) . \text { Then }, \]
\[y = x^2 + 7x + 2 ............. \left( 1 \right)\]
\[\text { Let the distance of a point } \left( x, \left( x^2 + 7x + 2 \right) \right) \text { from the line } y = 3x - 3\text { be S . Then,} \]
\[S = \left| \frac{- 3x + \left( x^2 + 7x + 2 \right) + 3}{\sqrt{10}} \right|\]
\[ \Rightarrow \frac{dS}{dt} = \frac{- 3 + 2x + 7}{\sqrt{10}}\]
\[\text { For maximum or minimum values of S, we must have }\]
\[\frac{dS}{dt} = 0\]
\[ \Rightarrow \frac{- 3 + 2x + 7}{\sqrt{10}} = 0\]
\[ \Rightarrow 2x = - 4\]
\[ \Rightarrow x = - 2\]
\[\text { Now }, \]
\[\frac{d^2 S}{d t^2} = \frac{2}{\sqrt{10}} > 0\]
\[\text { So, the nearest point is} \left( x, \left( x^2 + 7x + 2 \right) \right) . \]
\[ \Rightarrow \left( - 2, 4 - 14 + 2 \right)\]
\[ \Rightarrow \left( - 2, - 8 \right)\]
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