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?

Step-by-step solution

Let equation of common tangent is ( y = m x + 1 ~m ) ( | 0+0+ 1 ~m 1+ m ² |= 1 2 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 ( x² 1 + y² 1 / 2 =1 ) (A) ( e = 1- 1 2 = 1 2 ) and ( LR = 2 ~b ² a =1 ) (C) Area (2 _ 1 / 2 ¹ 1 2 1- x ² dx = 2 [ x 2 1- x ² + 1 2 ⁻¹ x ]_ 1 / 2 ¹ ) (= 2 [ 4 - ( 1 4 + 8 ) ]= 2 ( 8 - 1 4 ]= -2 4 2 ) Correct answers are (A) and (C).