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MHT CET202617 April 2026Evening ShiftMathematicsApplication of DerivativesActual

The coordinates of the points on the curve 4y = x^2 that are nearest to the point (0,5) are ...

Options

  1. A(-2 3 , 3)
  2. B(2 3 , -3)
  3. C(3, 2 3 )
  4. D(2 3 , 2)

Correct answer

A. (-2 3 , 3)

Step-by-step solution

Let the point on the curve be (x, y) . The square of the distance S between (x, y) and (0, 5) is given by: S = x^2 + (y - 5)^2 Since the point lies on the curve 4y = x^2 , we substitute x^2 = 4y into the equation for S : S = 4y + (y - 5)^2 S = y^2 - 6y + 25 To find the minimum distance, we differentiate S with respect to y and equate it to zero: dS dy = 2y - 6 = 0 y = 3 Since d^2S dy^2 = 2 > 0 , the distance is minimum at y = 3 . Substituting y = 3 in x^2 = 4y , we get: x^2 = 12 x = 2 3 The points on the curve near

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