Let a circle of radius 4 be concentric to the ellipse 15 x ^ 2 + 19...

Given,

$15{x}^{2} + 19{y}^{2} = 285$

$⇒ \dfrac{{x}^{2}}{19} + \dfrac{{y}^{2}}{15} = 1$

Now plotting the diagram with given circle of radius $4$ we get,

Now, let the equation of tangent of ellipse be $y = mx \pm \sqrt{19{m}^{2} + 15}$ \left\{\text{as}{c}^{2} = {a}^{2}{m}^{2} + {b}^{2}\right\}$</p><p>If it is tangent to circle${x}^{2} + {y}^{2} = 16$then using the formula,${c}^{2} = {r}^{2}\left(1 + {m}^{2}\right)$we get,</p><p>$\dfrac{\sqrt{19{m}^{2} + 15}}{\sqrt{1 + {m}^{2}}} = 4$</p><p>$⇒ 19{m}^{2} + 15 = 16 + 16{m}^{2}$</p><p>$⇒ 3{m}^{2} = 1$</p><p>$⇒ m = \pm \dfrac{1}{\sqrt{3}}$</p><p>$⇒ \theta = {30}^{°} with $x -$axis,

Hence, Angle made by tangent with minor axis i.e., with $y$ axis is $\dfrac{\pi}{3}$