### Instruction: An ellipse with equation
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2  +y^2 = 1.$  Then the smallest possible area of the ellipse can be expressed in the form $k \pi.$  Find $k.$.### Response: I notice that the circles are tangent to the ellipse at the points $(\pm 2, 0),$ so these are the endpoints of the major axis of the ellipse.
That means the major axis has length $4$ and the semi-major axis has length $a = 2.$
To minimize the area of the ellipse, I want to make the minor axis as small as possible, while still containing the circles.
I imagine shrinking the minor axis until it is just long enough to touch the circles at their highest and lowest points, which are $(\pm 1, \pm 1).$
These points satisfy the equation of the ellipse, so I can plug them in and solve for $b.$
I get $\frac{1}{4} + \frac{1}{b^2} = 1,$ which simplifies to $b^2 = \frac{4}{3}.$
The area of the ellipse is $\pi ab,$ so the smallest possible area is $\pi (2) (\sqrt{\frac{4}{3}}) = \frac{4 \sqrt{3}}{3} \pi.$