Explanation: The equation of the given curve:
$\frac{{\mathit{x}}^2}{9}+\frac{{\mathit{y}}^2}{25}=1$
On differentiating both sides w.r.t. $\mathit{x}$ , we get
$\frac{2\mathit{x}}{9}+\frac{2\mathit{y}}{25}\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=1\cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(\mathit{i}\right)$
$\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}=\frac{-25\mathit{x}}{9\mathit{y}}$
Since, tangent is parallel to $\mathit{X}$ -axis, then the slope of the tangent is zero.
$\therefore \frac{-25}{9}\frac{\mathit{x}}{\mathit{y}}=0$ , which is possible if $\mathit{x}=0$
Put $\mathit{x}=0$ in eq (i), we get
$\frac{{\mathit{y}}^2}{25}=1\Rightarrow {\mathit{y}}^2=25\Rightarrow \mathit{y}=\pm 5$
Hence, required points are $\left(0,\pm 5\right)$ .