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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 the 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 1162(π2)
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Solution

The correct options are
B The area of the region bounded by the ellipse between the lines x=12 and x=1 is 142(π2)
D For the ellipse, the eccentricity is 12 and the length of the latus rectum is 1
Let equation of common tangent is y=mx+1m
∣ ∣ ∣0+0+1m1+m2∣ ∣ ∣=12m4+m22=0m=±1
Equation of common tangents are y=x+1 and y=x+1
point Q is (1,0)
Equation of ellipse is x21+y21/2=1
(A) e=112=12 and LR=2b2a=1
(C) Area 2.11/212.1x2dx=2[x21x2+12sin1x]11/2
=2[π4(14+π8)]=2(π814]=π242
Correct answers are (A) and (C).
827531_903688_ans_8c12a932322c40c99b33957c3cb6b82c.png

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