The line 2x + y = 3 interesects the ellipse 4x^ 2 + y^ 2 = 5 at two...

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

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

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