The point on the ellipse 16 x 2 + 9 y 2 = 400 , where the ordinate decreases at the same rate at which the abscissa increases is a , b , then a + 3 b can be

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Aicharya · Accepted Answer

Differentiating the equation of ellipse, we get

32 x \frac{d x}{d t} + 18 y \frac{d y}{d t} = 0

\Rightarrow 32 x \frac{d x}{d t} = - 18 y \frac{d y}{d t}

As

\frac{d x}{d t} = - \frac{d y}{d t}

∴ 32 x = 18 y

\Rightarrow 16 x = 9 y

Putting in the equation of the ellipse, we get,

16 x^{2} + \frac{256}{81} \cdot 9 x^{2} = 400

\Rightarrow x^{2} + \frac{16}{9} x^{2} = 25

\Rightarrow x = \pm 3

\Rightarrow y = \pm \frac{16}{3}

∴ a = 3, b = \frac{16}{3}

\Rightarrow a + 3 b = 3 + 16 = 19

The point on the ellipse 16 x 2 + 9 y 2 = 400 , where the ordinate decreases at the same rate at which the…

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