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JEE Main201910 Jan 2019Evening ShiftMathematicsApplication of DerivativesActual

A helicopter is flying along the curve given by y - x 3 2 = 7 , x ≥ 0 . A soldier positioned at the point 1 2 , 7 , who wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:

Options

  1. A1 6 7 3
  2. B1 2
  3. C1 3 7 3
  4. D5 6

Correct answer

A. 1 6 7 3

Step-by-step solution

We have y = 7 + x 3 2 ⇒ d y d x = 3 2 x Let the point on curve be P x 1 ,   7 + x 1 3 2 and given point be A 1 2 ,   7 . For nearest point normal at P passes through A . So, the slope of line A P = Slope of normal at P ⇒ x 1 3 2 x 1 - 1 2 = - d x d y x 1 , y 1   = - 2 3 x 1 ⇒ 3 x 1 2 = 1 - 2 x 1 ⇒ 3 x 1 2 + 2 x 1 - 1 = 0 ⇒ x 1 + 1 3 x 1 - 1 = 0 ⇒ x 1 = 1 3   ( x 1 = - 1 , is not possible as x 1 > 0 ) Hence, point P is 1 3 ,   7 + 1 3 3 . So, A P = 1

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