Consider two straight lines, each of which is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 2 }$ 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 $\sqrt { 2 }$, then which of the following statement(s) is (are) TRUE? (A) For the ellipse, the eccentricity is $\frac { 1 } { \sqrt { 2 } }$ and the length of the latus rectum is 1 (B) For the ellipse, the eccentricity is $\frac { 1 } { 2 }$ and the length of the latus rectum is $\frac { 1 } { 2 }$ (C) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is $\frac { 1 } { 4 \sqrt { 2 } } ( \pi - 2 )$ (D) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is $$\frac { 1 } { 16 } ( \pi - 2 )$$
Consider two straight lines, each of which is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 2 }$ 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 $\sqrt { 2 }$, then which of the following statement(s) is (are) TRUE?\\
(A) For the ellipse, the eccentricity is $\frac { 1 } { \sqrt { 2 } }$ and the length of the latus rectum is 1\\
(B) For the ellipse, the eccentricity is $\frac { 1 } { 2 }$ and the length of the latus rectum is $\frac { 1 } { 2 }$\\
(C) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is $\frac { 1 } { 4 \sqrt { 2 } } ( \pi - 2 )$\\
(D) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is
$$\frac { 1 } { 16 } ( \pi - 2 )$$