Question:
An Apache helicopter o the enemy is flying along the curve given by y = x2 + 7. A soldier placed at (3, 7) wants to shoot down the
helicopter when it is nearest to him. Find the nearest distance between the soldier and helicopter.
Answer:
Let the point (x, y) be placed the current position of helicopter along the curve y = x2 + 7
Given, soldier is placed at the point (3, 7)
So, the distance between helicopter and soldier = D
Now, D2 = (x - 3)2 + (y - 7)2
=> D2 = (x - 3)2 + (x2 + 7 - 7)2
=> D2 = (x - 3)2 + x4
Differentiate w.r.t. x, we get
=> dD2 /dx = d(x - 3)2 /dx + dx4 /dx
=> dD2 /dx = 2(x - 3) + 4x3
For minimum D2 , we have
=> dD2 /dx = 0
=> 2(x - 3) + 4x3 = 0
=> 2(x - 3 + 2x3 ) = 0
=> 2x3 + x - 3 = 0
=> 2(x - 1) * (2x2 + 2x + 3) = 0
=> x = 1 and 2x2 + 2x + 3 = 0
Since there is no real roots for the equation 2x2 + 2x + 3 = 0
So, x = 1
d2 D2 /dx2 = 2 + 12x2
=> d2 D2 /dx2 = 2(6x2 + 1)
For x = 1
=> d2 D2 /dx2 = 2(6 * 1 + 1)
=> d2 D2 /dx2 = 14 > 0
Hence, at x = 1, distance is minimum.
Put x = 1 in the given equation, we get
y = 1 + 7 = 8
Therefore point (1, 8) is nearest to the point (3, 7)
Now, minimum ditance = √{(3 - 1)2 + (7 - 8)2 } = √5
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