Question

The points of the ellipse \(16 x^{2}+9 y^{2}=400\) at which the ordinate decreases at the same rate at which the abscissa increases is/are given by
(A) \(\left(3, \frac{16}{3}\right) \&\left(-3, \frac{-16}{3}\right)\)
(B) \(\left(3, \frac{-16}{3}\right) \&\left(-3, \frac{16}{3}\right)\)
(C) \(\left(\frac{1}{16}, \frac{1}{9}\right) \&\left(-\frac{1}{16},-\frac{1}{9}\right)\)
(D) \(\left(\frac{1}{16},-\frac{1}{9}\right) \&\left(-\frac{1}{16}, \frac{1}{9}\right)\)

Answer 1

Ans: (A)
 \(\frac{\frac{x^{2}}{25}}{\frac{y^{2}}{9}}=\frac{y}{\frac{400}{9}}\)
\(\left(5 \cos \theta, \frac{20}{3} \sin \theta\right)\)
\(x=5 \cos \theta, y=\frac{20}{3} \sin \theta\)
\(\frac{d x}{d \theta}=-5 \sin \theta, \frac{d y}{d \theta}=\frac{20}{3} \cos \theta\)
\(\frac{d x}{d \theta}=-\frac{d y}{d \theta}\)
\(-5 \sin \theta=-\frac{20}{3} \cos \theta\)
\(\tan \theta=4 / 3\)
\(\Rightarrow \cos \theta=3 / 5\) or \(-3 / 5\)
\(\sin \theta=4 / 5\) or \(-4 / 5\)
Points are \(\left(3, \frac{16}{3}\right)\) and \(\left(-3, \frac{-16}{3}\right)\)

Answer 2

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Answer 3

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