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# Find the Coordinates of a Point on the Parabola Y=X2+7x + 2 Which is Closest to the Strainght Line Y = 3x − 3 ? - CBSE (Commerce) Class 12 - Mathematics

#### 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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Solution Find the Coordinates of a Point on the Parabola Y=X2+7x + 2 Which is Closest to the Strainght Line Y = 3x − 3 ? Concept: Graph of Maxima and Minima.
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