Consider two lines which are common tangents to both the curves $x^{2} + y^{2} = \frac{1}{2}$ and $y^{2} = 4 x$ . Let these lines intersect at the point Q. Consider the ellipse with centre at the origin and whose semi – major axis is OQ. Length of its minor axis is $\sqrt{2}$ . Choose the correct statement(s) among the following.

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The eccentricity of the ellipse is $\frac{1}{\sqrt{2}}$ and length of its latusrectum is 1.

The area of the polygon formed by the foci and ends of minor axis is 1 sq.unit.

The area of the polygon formed by the foci and ends of minor axis is 2 sq.units.

The eccentricity of the ellipse is $\frac{1}{2}$ and length of its latusrectum is $\frac{1}{2}$ .

answer is A, D.

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Detailed Solution

Let $y = m x + c$ be the common tangent
$∴ C = \frac{1}{m}$ & $C^{2} = \frac{1}{2} (1 + m^{2})$
$\Rightarrow \frac{1}{m^{2}} = \frac{1}{2} (1 + m^{2}) \Rightarrow m = \pm 1$
$∴ C T_{s}$ are $y = x + 1$ and $y = - x - 1 \Rightarrow Q = (- 1, 0)$
For ellipse $a = 1, b = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$
$e = \sqrt{\frac{1 - \frac{1}{2}}{1}} = \frac{1}{\sqrt{2}}$ , LLR $= \frac{2. b^{2}}{a} = \frac{2. \frac{1}{2}}{1} = 1$
Area $= 4 (\frac{1}{2} . b . a . e) = 1 s q . u n i t$