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From any point on the ellipse x²/9 + y²/4 = 1, tangents are drawn to the circle x² + y² = 1, and the corresponding chord of contact always touches a conic C. Which of the following is correct?
- eccentricity of C is sqrt(5)/3
- minimum distance between C and the ellipse is 3/2
- maximum distance between C and the ellipse is 10/3
- product of eccentricities of C and the ellipse is 1
Correct answer: eccentricity of C is sqrt(5)/3
Solution
The chord of contact 3x cos t + 2y sin t = 1 envelopes 9x² + 4y² = 1, i.e. an ellipse with a² = 1/4 and b² = 1/9; its eccentricity is sqrt(1 - (1/9)/(1/4)) = sqrt(5)/3.
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