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Q.

Consider two straight lines, each of which is tangent to both the circle x2+y2=12 and the parabola y2=4xLet 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?

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a

For the ellipse, the eccentricity is 12 and the length of the latus rectum is 1

b

For the ellipse, the eccentricity is 12 and the length of the latus rectum is 12

c

The area of the region bounded by the ellipse between the lines x=12 and x=1 is142π−2

d

The area of the region bounded by the ellipse between the lines x=12 and x=1 is116π−2

answer is A.

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Detailed Solution

Let equation of common tangent is y=mx+1m0+0+1m1+m2=12⇒m4+m2−2=0⇒m=±1Equation of common tangents are y=x+1 and y=−x−1 Point Q is −1,0Equation of ellipse is x21+y21/2=1A) e=1+12=12 and LR=2b2a=1 Area 2. ∫1/2212·1+x2dx=2x21-x2912sin-1x121=2π4-14+π8=2π8-14=π-242

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Consider two straight lines, each of which is tangent to both the circle x2+y2=12 and the parabola y2=4xLet 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?