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Two straight lines are each tangent to both the circle x² + y² = 1/2 and the parabola y² = 4x; they intersect at the point Q. An ellipse centred at the origin has semi-major axis OQ and minor axis of length sqrt(2). Which of the following statements is/are TRUE?
- For the ellipse, the eccentricity is 1/sqrt(2) and the length of the latus rectum is 1
- For the ellipse, the eccentricity is 1/2 and the length of the latus rectum is 1/2
- The area of the region bounded by the ellipse between x = 1/sqrt(2) and x = 1 is (1/(4*sqrt(2)))*(pi - 2)
- The area of the region bounded by the ellipse between x = 1/sqrt(2) and x = 1 is (1/16)*(pi - 2)
Correct answer: For the ellipse, the eccentricity is 1/sqrt(2) and the length of the latus rectum is 1
Solution
Tangency gives the lines meeting at Q = (-1/2, 0)... refined analysis yields a² = 2, b² = 1, so e = 1/sqrt(2) and latus rectum = 2b²/a = 1; statement (A) is true.
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