Ellipse
Common Tangent to Ellipse and Concentric Circle
nta_pyq_2023_apr
Grade 11
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
Step-by-Step Solution
Key Concept: Ellipse: $\frac{x^2}{19}+\frac{y^2}{15}=1$, $a^2=19$, $b^2=15$. Tangent $y=mx\pm\sqrt{19m^2+15}$. For tangency to circle $x^2+y^2=16$: $\frac{19m^2+15}{1+m^2}=16$.
$m=\frac{1}{\sqrt{3}}$. Angle with minor axis $=60°=\frac{\pi}{3}$.
Correct Answer: 1