An energy DRONE is flying along the curve y = x² + 7. A soldier is placed at (3, 7). The nearest distance of the DRONE from soldier’s position is

\(\sqrt { 5}\)

Let P(x, y) be position of the DRONE and the soldier is placed at A(3, 7).

$AP=\sqrt{(x-3)^2+(y-7)^2}$ .....(i)

$y = x^2 + 7$ .....(ii)

since point lies on the curve

$S=\sqrt{(x-3)^2+(x^2+7-7)^2}$

$S=\sqrt{x^4+x^2-6x+9}$

when distance is maximum/minimum

$\frac{dS}{dx}=0⇒\frac{(4x^3+2x-6)}{\sqrt{(x^4+x^2-6x+9)}}=0$

$⇒4x^3+2x-6=0$

$⇒2x^3+x-3=0$

$⇒(x-1) × (2x^2+2x+3)$

The solution to above equation are

$x = 1, -0.5 ± i × 0.5\sqrt{5}$

Since solution cannot have any complex roots.

Hence, x = 1 is abscissa of the nearest point to the soldier.

from eq. (i) we get,

$y  =1^2+7⇒y=8$

Nearest point is (1, 8)

nearest distance $S = \sqrt{(1-3)^2+(8-7)^2}$

$S = \sqrt{(-2)^2+(1)^2}⇒S=\sqrt{4+1}$

$\Rightarrow S=\sqrt{5}$