Q.
Two concentric ellipses are such that the foci of one are on the other and their major axes are equal. Let e and e' be their eccentricities. Then,
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a
the quadrilateral formed by joining the foci of the two ellipses is a parallelogram
b
the angle θ between their axes is given by θ=cos−11e2+1e′2−1e2e′2
c
If e2 + e'2 = 1, then the angle between the axes of the two ellipses is 90°
d
none of these
answer is A.
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Detailed Solution
Clearly, O is the midpoint of SS', and HH'.Therefore, the diagonals of quadrilateral HSH'S' bisect each other. So it is a parallelogram.Let H'OH = 2r. Then, OH = r = ae'.The point H lies onx2a2+y2b2=1( Suppose )∴r2cos2θa2+r2sin2θb2=1e2cos2θ+e2sin2θ1−e2=1 ∵b2=a21−e2 or e2cos2θ−e2cos2θ1−e2=1−e21−e2or cos2θ=1e2+1e′2−1e2e′2 or θ=cos−11e2+1e2−1e2e′2 For θ=90∘, e2+e2e2e2=1e2e2 or e2+e'2=1
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