1. Find the ellipse parameters

The ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ with eccentricity $e=\frac{1}{2}$ and foci $(\pm 2,0)$.

For a standard ellipse with major axis along the $x$-axis, $c=ae,$ where foci are $(\pm c,0)$.

Given $c=2$ and $e=\tfrac12$, $a=\frac{c}{e}=\frac{2}{1/2}=4.$

Now, $b^2=a^2(1-e^2)=16\left(1-\frac14\right)=16\cdot \frac34=12,$ so $b=2\sqrt{3}.$

Thus the ellipse is $\frac{x^2}{16}+\frac{y^2}{12}=1.$

2. Find the minimum-area circle enclosing the ellipse

The ellipse is centered at the origin, with semi-major axis $a=4$ and semi-minor axis $b=2\sqrt3<4$.

The smallest circle centered at the origin enclosing the ellipse has radius equal to the maximum distance of a point on the ellipse from the origin, which is $a=4$.

Hence the circle $C$ is $x^2+y^2=16.$

3. Interpret the triangle condition

The point where the ellipse meets the negative $y$-axis is $(0,-b)=(0,-2\sqrt3).$

The side $QR$:

• has length $2a=8$,
• is parallel to the major axis of the ellipse, i.e. parallel to the $x$-axis,
• contains $(0,-2\sqrt3)$.

So $QR$ lies on the horizontal line $y=-2\sqrt3.$

Since its length is $8$, its exact position on this line does not matter for area once the base is fixed; the area depends only on the perpendicular distance from $P$ to this line.

4. Write area in terms of point $P$ on circle $C$

Let $P=(x,y)$ be any point on $x^2+y^2=16.$

Area of triangle $PQR$ is $\Delta=\frac12 \times QR \times \text{distance from }P\text{ to line }QR.$

Since $QR=8$ and $QR$ is the line $y=-2\sqrt3$, $\Delta=\frac12\cdot 8\cdot |y+2\sqrt3|=4|y+2\sqrt3|.$

Because $P$ lies on the circle $x^2+y^2=16$, we have $-4\le y\le 4.$

We need to maximize $|y+2\sqrt3|.$

5. Maximize the distance

Since $2\sqrt3\approx 3.464$, $y+2\sqrt3\in[-4+2\sqrt3,\;4+2\sqrt3].$

Now, $|-4+2\sqrt3|=4-2\sqrt3,$ while $|4+2\sqrt3|=4+2\sqrt3.$

Clearly the maximum occurs at $y=4,$ that is, when $P=(0,4)$.

So the maximum area is $\Delta_{\max}=4(4+2\sqrt3)=16+8\sqrt3=8(2+\sqrt3).$

6. Check options

$8(2+\sqrt3)$ corresponds to Option B.

7. Compare with stored correct answer

Stored correct answer: B

Our derived answer: B

So they agree.