A helicopter is flying along the curve y= x2+2 A soldier is placed at the point (3,2) Find the - Maths - Application of Derivatives

Question

A helicopter is flying along the curve y= x2+2 A soldier is
placed at the point (3,2) Find the nearest distance between the
soldier and the helicopter (2010Sp)

Ankush Jain · Accepted Answer

Let the point (x, y) be the current position of helicopter along the curve (y = x2 + 2) Given soldier is placed at the point (3,2) Therefore, distance between helicopter and soldier = D $\sqrt{(x - 3)^{2} + (y - 2)^{2}}$ Now, we know that, if D is minimum then D2 is also minimum So, D2 =

$\frac{d D^{2}}{d x} = 2 (x - 3) + 4 (x)^{3} = 2 (2 x^{3} + x - 3)$ For minimum D2, we have $\frac{d D^{2}}{d x} = 0 \Rightarrow 2 (2 x^{3} + x - 3) = 0 \Rightarrow (2 x^{3} + x - 3) = 0 \Rightarrow 2 (x - 1) (2 x^{2} + 2 x + 3) = 0 \Rightarrow (x - 1) = 0 o r (2 x^{2} + 2 x + 3) = 0$ â‡’ x = 1 as there are no real rots for the equation $(2 x^{2} + 2 x + 3) = 0$ Again, $\frac{d^{2} D^{2}}{d x^{2}} = 2 (6 x^{2} + 1) N o w, f o r x = 1, \frac{d^{2} D^{2}}{d x^{2}} > 0$ Clearly, for x = 1, distance is minimum On putting value of x in the given curve , we get y = (1)2 + 2 = 3 Therefore, point (1,3) is nearest to point (3,2) Hence, minimum distance = $\sqrt{(3 - 1)^{2} + (2 - 3)^{2}} = \sqrt{5}$

A helicopter is flying along the curve y= x2+2.A soldier is placed at the point (3,2). Find the nearest distance between the soldier and the helicopter.(2010Sp)