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 $\frac{1}{\sqrt{2}}$
• B. For the ellipse, the eccentricity is $\frac{1}{2}$
• C. For the ellipse, the length of the latus rectum is $1$ unit
• D. For the ellipse, the length of the latus rectum is $\frac{1}{2}$

Answer 1

Answer: (A), (C)

The correct options are
A For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$
C For the ellipse, the length of the latus rectum is $1$ unit

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,
$OQ=a$
$\therefore a=1\text{and}b=\frac{1}{\sqrt{2}}$
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 $=\frac{2b^2}{a}=1$ unit