Given that, the ordinate decreases at the same rate at which the abscissa increases, therefore
$$
\frac{d y}{d t}=-\frac{d x}{d t}
$$
Also, equation of ellipse
$$
16 x^{2}+9 y^{2}=400
$$
On differentiating wrt. $t$, we get
$16 \times 2 x \frac{d x}{d t}+9 \times 2 y \frac{d y}{d t}=0$
$\Rightarrow \quad 16 x \frac{d x}{d t}+9 y \frac{d y}{d t}=0$
$\Rightarrow \quad 9 y \frac{d y}{d t}=-16 x \frac{d x}{d t}$
$\Rightarrow \quad 9 y \frac{d y}{d t}=-16 x\left(-\frac{d y}{d t}\right) \quad$ [using $\left.\mathrm{Eq.}(i)\right]$
$\Rightarrow \quad 9 y=16 x \Rightarrow y=\frac{16}{9} x$
$16 x^{2}+9\left(\frac{16}{9} x\right)^{2}=400$
$\Rightarrow \quad 16 x^{2}+\frac{(16)^{2}}{9} x^{2}=400$
$\Rightarrow \quad 16 x\left[x+\frac{16}{9} x\right]=400$
$\Rightarrow \quad 16 x\left[\frac{25}{9} x\right]=400 \Rightarrow x^{2}=\frac{400 \times 9}{25 \times 16}$
$\Rightarrow \quad x^{2}=9 \Rightarrow x=\pm 3$
$\therefore$ Required points are $\left(3, \frac{16}{3}\right)$ and $\left(-3-\frac{16}{3}\right)$