COMEDK · Maths · 27. Application of Derivatives

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

A \(\left(3, \dfrac{16}{3}\right) \text { and }\left(-3, \dfrac{16}{3}\right)\)
B \(\left(-3, \dfrac{16}{3}\right) \text { and }\left(3,-\dfrac{16}{3}\right)\)
C \(\left(3, \dfrac{16}{3}\right) \text { and }\left(-3,-\dfrac{16}{3}\right)\)
D \(\left(-3,-\dfrac{16}{3}\right) \text { and }\left(-3, \dfrac{16}{3}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\left(3, \dfrac{16}{3}\right) \text { and }\left(-3,-\dfrac{16}{3}\right)\)

Step-by-step Solution

Detailed explanation

The given equation of the ellipse is \(16x^2 + 9y^2 = 400\). Differentiating both sides with respect to time \(t\), we get \(32x \dfrac{dx}{dt} + 18y \dfrac{dy}{dt} = 0\). The problem states that the ordinate decreases at the same rate at which the abscissa increases, which…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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