Consider two straight lines, each of which is tangent to both the circle $x^2+y^2=\frac{1}{2}$ and the parabola $y^2=4x$ . 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 $\sqrt{2}$, then the which of the following statement(s) is (are) TRUE?

Let equation of common tangent is \(\mathrm{y}=\mathrm{m} \mathrm{x}+\frac{1}{\mathrm{~m}}\)

\(\therefore\left|\frac{0+0+\frac{1}{\mathrm{~m}}}{\sqrt{1+\mathrm{m}^{2}}}\right|=\frac{1}{\sqrt{2}} \Rightarrow \mathrm{m}^{4}+\mathrm{m}^{2}-2=0 \Rightarrow \mathrm{m}=\pm 1\) Equation of common tangents are \(\mathrm{y}=\mathrm{x}+1\) and \(\mathrm{y}=-\mathrm{x}+1\)

point \(Q\) is \((-1,0)\)

\(\therefore\) Equation of ellipse is \(\frac{x^{2}}{1}+\frac{y^{2}}{1 / 2}=1\)

(A) \(\mathrm{e}=\sqrt{1-\frac{1}{2}}=\frac{1}{\sqrt{2}}\) and \(\mathrm{LR}=\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=1\)

(C) Area \(2 \cdot \int_{1 / \sqrt{2}}^{1} \frac{1}{\sqrt{2}} \cdot \sqrt{1-\mathrm{x}^{2}} \mathrm{dx}=\sqrt{2}\left[\frac{\mathrm{x}}{2} \sqrt{1-\mathrm{x}^{2}}+\frac{1}{2} \sin ^{-1} \mathrm{x}\right]_{1 / \sqrt{2}}^{1}\) \(=\sqrt{2}\left[\frac{\pi}{4}-\left(\frac{1}{4}+\frac{\pi}{8}\right)\right]=\sqrt{2}\left(\frac{\pi}{8}-\frac{1}{4}\right]=\frac{\pi-2}{4 \sqrt{2}}\)

Correct answers are (A) and (C).