The acute angle between the pair of tangents drawn to the ellipse 2...

Let equation of tangents to the ellipse $\dfrac{{x}^{2}}{\dfrac{5}{2}} + \dfrac{{y}^{2}}{\dfrac{5}{3}} = 1$ be $y = mx \pm \sqrt{\dfrac{5}{2}{m}^{2} + \dfrac{5}{3}}$

Since the tangents pass through \left(1, 3\right)$, so </p><p>$3 = m \pm \sqrt{\dfrac{15{m}^{2} + 10}{6}}$</p><p>$⇒ 9{m}^{2} + 36m - 44 = 0$</p><p>Let$\theta$be the angle between the tangents, so </p><p>$tan\theta = \left|\dfrac{{m}_{1} - {m}_{2}}{1 + {m}_{1}{m}_{2}}\right|

i.e. $tan\theta = \left|\dfrac{\sqrt{{4}^{2} + 4\left(\dfrac{44}{9}\right)}}{1 - \dfrac{44}{9}}\right| = \dfrac{24\sqrt{5}}{35}$

$⇒ \theta =$\tan^{- 1}\left(\dfrac{24}{7\sqrt{5}}\right)