The coordinates of the point on the ellipse `16 x ^(2) + 9y ^(2) = 400` where the ordinate decreases at the same rate at which abscissa increases are :

Options

A. `(3, 16//3)`
B. `(-3, 16//3)`
C. `(3, 16//3)`
D. `(3, -3)`

Answer

A

Explanation

We have,
`16x^(2)+9y^(2)=400`
`implies 32x(dx)/(dt)+18 y (dy)/(dt)=0 " [Differentiating w.r.t. t]"`
`implies 32x(dx)/(dt)-18y(dx)/(dt)=0 " "[because-(dy)/(dt)=(dx)/(dt)("given")]`
`implies 16x-9y=0`
`impliesy=(16x)/(9)`
`therefore 16x^(2)+9y^(2)=400 implies 16x^(2)+(256)/(9)x^(2)=400impliesx=pm3`
Now, `y=(16)/(9)x and x=pm3impliesy= pm(16)/(3)`
Hence, the required points are `(3, 16//3)` and `(-3, -16//3)`