Question

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 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 $\frac{1}{16}(\pi -2)$

Answer 1

Answer: (A), (C)

The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{4\sqrt{2}}(\pi -2)$

For parabola, equation of tangent in terms of slope is $y=mx+\frac{1}{m}$
For circle, equation of tangent in terms of slope is $y=mx\pm \frac{1}{\sqrt{2}}\sqrt{1+m^2}$
$\frac{1}{m}=\pm \frac{1}{\sqrt{2}}\sqrt{1+m^2}$
$\Rightarrow m=\pm 1$
Therefore, equation of common tangents are $y=x+1,y=-x-1$
$Q$ is the intersection of the two tangents.
$\therefore$ Coordinates of $Q$ is $(-1,0)$.

For ellipse,
$a=1\text{and}b=\frac{1}{\sqrt{2}}$
Equation of ellipse is $\frac{x^2}{1}+\frac{y^2}{1/2}=1$
Eccentricity, $e=\sqrt{1-\frac{1/2}{1}}=\frac{1}{\sqrt{2}}$
Length of latus rectum $=\frac{2b^2}{a}=1$


$\text{Area}=2\underset{1/\sqrt{2}}{\overset{1}{\int }}\frac{1}{\sqrt{2}}\sqrt{1-x^2}dx$
$=\sqrt{2}{[\frac{x}{2}\sqrt{1-x^2}+\frac{1}{2}{\sin }^{-1}x]}_{1/\sqrt{2}}^1=\frac{\pi -2}{4\sqrt{2}}$