Consider two straight lines, each of which is tangent to both the circle x 2 + y 2 = 1 2 and the parabola y 2 = 4 x . Let these lines intersect at the point Q . Consider the ellipse whose center is at

Question

Consider two straight lines, each of which is tangent to both the circle x 2 + y 2 = 1 2 and the parabola y 2 = 4 x . 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 O Q . If the length of the minor axis of this ellipse is 2 , then the which of the following statement(s) is (are) TRUE?

TestHub · Accepted Answer

Let equation of common tangent is y = mx + m 1 ​ ∴ ​ 1 + m 2 ​ 0 + 0 + m 1 ​ ​ ​ = 2 ​ 1 ​ ⇒ m 4 + m 2 − 2 = 0 ⇒ m = ± 1 Equation of common tangents are y = x + 1 and y = − x + 1 point Q is ( − 1 , 0 ) ∴ Equation of ellipse is 1 x 2 ​ + 1/2 y 2 ​ = 1 (A) e = 1 − 2 1 ​ ​ = 2 ​ 1 ​ and LR = a 2 b 2 ​ = 1 (C) Area 2 ⋅ ∫ 1/ 2 ​ 1 ​ 2 ​ 1 ​ ⋅ 1 − x 2 ​ dx = 2 ​ [ 2 x ​ 1 − x 2 ​ + 2 1 ​ sin − 1 x ] 1/ 2 ​ 1 ​ = 2 ​ [ 4 π ​ − ( 4 1 ​ + 8 π ​ ) ] = 2 ​ ( 8 π ​ − 4 1 ​ ] = 4 2 ​ π − 2 ​ Correct answers are (A) and (C).

Mathematics - Ellipse Question with Solution | TestHub

Options:(select one or more)

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