Let the required point be $\left(\mathrm{h},\mathrm{k}\right).$

Now from this point, the equation of chord of contact to the ellipse is $\mathrm{T}=0$

$\Rightarrow 4\mathrm{hx}+\mathrm{ky}=5$ which is same as $2\mathrm{x}+\mathrm{y}=3$
$\Rightarrow \frac{4\mathrm{h}}{2}=\frac{\mathrm{k}}{1}=\frac{5}{3}\Rightarrow \mathrm{h}=\frac{5}{6}, \mathrm{k}=\frac{5}{3}$
$\Rightarrow$ Point of intersection of the tangents is $\left(\frac{5}{6},\frac{5}{3}\right)$