A point on the ellipse $4x^2+9y^2=36$ where the tangent is equally inclined to the coordinate axes is

$8x+18y\frac{dy}{dx}=0$

$9y{y}^1=-4x$

$y^1=\frac{-4x}{9y}$

Tangent is equals inclined to the axes

$\frac{-4x}{9y}=1,\frac{-4x}{9y}=-1$

$4x=-9y,4x=9y$

$y=\frac{4x}{9},y=\frac{-4x}{9}$

$4x^2+9\cdot \frac{16x^2}{81}=36$

$4x^2+\frac{16x^2}{9}=36$

$52x^2=324$

$x^2=\frac{324}{52}$

$$\begin{gathered} =\frac{162}{26}=\frac{81}{13} \\ x=\pm \frac{9}{\sqrt{13}};y=\mp \frac{4}{\sqrt{13}} \end{gathered}$$