Let a circle of radius 4 be concentric to the ellipse 15x^2+19y^2=285.

Question

Let a circle of radius $4$ be concentric to the ellipse $15x^2+19y^2=285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle:

Accepted Answer

Ellipse: $\dfrac{x^2}{19}+\dfrac{y^2}{15}=1$. Tangent: $y=mx\pm\sqrt{19m^2+15}$, tangent to the concentric circle of radius $4$:

$$\frac{\sqrt{19m^2+15}}{\sqrt{1+m^2}}=4\ \Longrightarrow\ 3m^2=1\ \Longrightarrow\ m=\pm\frac1{\sqrt3}.$$

This gives $\theta=\dfrac\pi6$ with the $x$-axis (major axis), so the angle with the minor axis ($y$-axis) is $\dfrac\pi2-\dfrac\pi6=\dfrac\pi3.$

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