A jet of enemy is flying along the curve $y = x^{2} + 2$ and a soldier is placed at the point (3,2).Find the minimum distance between the soldier and the jet.

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Solution

Let P(x,y) be the position of the jet and the soldier is placed at A(3,2).
$\Rightarrow A P = \sqrt{(x - 3)^{2} + (y - 2)^{2}}$ ...(i)
As y= $(x^{2} + 2 \Rightarrow x^{2} = y - 2$ ...(ii)
$\Rightarrow A P^{2} = (x - 3)^{2} + x^{4} = z$ (Say)
$\Rightarrow \frac{d z}{d x} = 2 (x - 3) + 4 x^{3} and \frac{d^{2} z}{d x^{2}} = 2 + 12 x^{2}$
For local points of maxima/minima, $\frac{d z}{d x} = 0 \Rightarrow 2 (x - 3) + 4 x^{3} = 0 \Rightarrow x = 1$
And $\frac{d^{2} z}{d x^{2}} (a t x = 1) = 14 > 0$
$∴$ z is minimum when x=1,y=1+2=3
Also minimum distance = $\sqrt{(1 - 3)^{2} + (3 - 2)^{2}} = \sqrt{5}$ units.

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A jet of enemy is flying along the curve y=x2+2 and a soldier is placed at the point (3,2).Find the minimum distance between the soldier and the jet.

A jet of enemy is flying along the curve $y = x^{2} + 2$ and a soldier is placed at the point (3,2).Find the minimum distance between the soldier and the jet.