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Angle Between Tangents from the Director Circle | JEE

JEE Maths question with a full step-by-step solution.

Question
Tangents are drawn from any point on the circle x2+y2=41x^2+y^2=41 to the ellipse x225+y216=1\dfrac{x^2}{25}+\dfrac{y^2}{16}=1. The angle between the two tangents is:
Aπ4\dfrac{\pi}{4}
Bπ3\dfrac{\pi}{3}
Cπ6\dfrac{\pi}{6}
Dπ2\dfrac{\pi}{2}correct
Solution
Step 1: The director circle of x225+y216=1\dfrac{x^2}{25}+\dfrac{y^2}{16}=1 is x2+y2=25+16=41x^2+y^2=25+16=41. Step 2: The given circle x2+y2=41x^2+y^2=41 is exactly this director circle. From any point on the director circle, the two tangents to the ellipse are perpendicular. ∴\therefore the angle between the tangents is π2\dfrac{\pi}{2}. Correct answer: (4)
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