Let $A D$ and $B C$ be two vertical poles at $A$ and $b$ respectively on a horizontal ground. If $A D=8 \mathrm{~m}, B C=11 \mathrm{~m}$ and $A B=10 \mathrm{~m}$; then the distance (in meters) of a point $M$ on $A B$ from the point $A$ such that $M D^{2}+M C^{2}$ is minimum is ______.

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Application of Derivatives

Let

A D

and

B C

be two vertical poles at

A

and

b

respectively on a horizontal ground. If

A D = 8 m, B C = 11 m

and

A B = 10 m

; then the distance (in meters) of a point

M

A B

from the point

A

such that

M D^{2} + M C^{2}

is minimum is ______.

Answer: 5

Let

A M = x

, so

M B = 10 - x

. The poles are vertical, so

M D^{2} = x^{2} + A D^{2} = x^{2} + 64

and

M C^{2} = (10 - x)^{2} + B C^{2} = (10 - x)^{2} + 121

. Adding,

M D^{2} + M C^{2} = x^{2} + (10 - x)^{2} + 185 = 2 x^{2} - 20 x + 285 = 2 (x - 5)^{2} + 235

, which is least when

x = 5

. The point

M

therefore lies

5 m

from

A

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Integer answer