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COMEDK2024MathematicsApplication of DerivativesActual

The points on the ellipse 16 x^2+9 y^2=400 at which the ordinate decreases at the same rate at which the abscissa increases are

Options

  1. A(3, 16 3 ) and (-3, 16 3 )
  2. B(-3, 16 3 ) and (3,- 16 3 )
  3. C(3, 16 3 ) and (-3,- 16 3 )
  4. D(-3,- 16 3 ) and (-3, 16 3 )

Correct answer

C. (3, 16 3 ) and (-3,- 16 3 )

Step-by-step solution

The given equation of the ellipse is 16x^2 + 9y^2 = 400 . Differentiating both sides with respect to time t , we get 32x dx dt + 18y dy dt = 0 . The problem states that the ordinate decreases at the same rate at which the abscissa increases, which means dy dt = - dx dt . Substituting this into the differentiated equation: 32x dx dt + 18y (- dx dt ) = 0 . Assuming dx dt 0 , we have 32x - 18y = 0 , which simplifies to 16x = 9y , or y = 16x 9 . Substitute y = 16x 9 into the original ellipse equation: 16x^2 + 9 ( 16x 9

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