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Question

Consider two straight lines, each of which is tangent to both the circle x2+y2=12 and the parabola y2=4x. Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0,0) and whose semi-major axis is OQ. If the length of the minor axis of this ellipse is 2, then which of the following statement(s) is (are) TRUE?

A
For the ellipse, the eccentricity is 12 and the length of the latus rectum is 1
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B
For the ellipse, the eccentricity is 12 and the length of the latus rectum is 12
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C
The area of the region bounded by the ellipse between the lines x=12 and x=1 is 142(π2)
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D
The area of the region bounded by the ellipse between the lines x=12 and x=1 is 116(π2)
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Solution

The correct option is C The area of the region bounded by the ellipse between the lines x=12 and x=1 is 142(π2)
For parabola, equation of tangent in terms of slope is y=mx+1m
For circle, equation of tangent in terms of slope is y=mx±121+m2
1m=±121+m2
m=±1
Therefore, equation of common tangents are y=x+1, y=x1
Q is the intersection of the two tangents.
Coordinates of Q is (1,0).

For ellipse,
a=1 and b=12
Equation of ellipse is x21+y21/2=1
Eccentricity, e=11/21=12
Length of latus rectum =2b2a=1


Area=211/2121x2 dx
=2[x21x2+12sin1x]11/2=π242

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