On the ellipse 4x² + 9y² = 1, the points at which the tangents are parallel to the line 8x = 9y are

$Given ellipse \frac{x^2}{{\displaystyle \frac{1}{4}}}+\frac{y^2}{{\displaystyle \frac{1}{9}}}=1$$\to \left(1\right)$

The line parallel to 8x -9y =  0  be 8x- 9y+ k= 0
If it is a tangent to (1), then $n^2=a^2{l}^2+b^2{m}^2$

$\Rightarrow {\mathrm{k}}^2=25\therefore$The tangent are $8\mathrm{x}-9\mathrm{y}\pm 5=0$

Now points of contact are $(\frac{-a^{2}l}{n},\frac{-b^{2}m}{n})$
$\left(-\frac{2}{5},\frac{1}{5}\right)\left(\mathrm{or}\right)\left(\frac{2}{5},-\frac{1}{5}\right)$