An ellipse is tangent to both the $x$-axis and the $y$-axis, and its foci are at $(2, -3)$ and $(2, -8).$ Find the length of the major axis.
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The major axis clearly lies on the line x = 2, with the center at (2,-11/2).
So a = 11/2 and the major axis has length 11.
That's correct! Since the ellipse is tangent to both the $x$-axis and the $y$-axis, its center lies on the line $x=2$ and $y=-\frac{11}{2}.$ Therefore, the distance from the center to each focus is $c=\frac{5}{2}.$ The length of the major axis is $2a,$ where $a$ is the distance from the center to each vertex. Using the Pythagorean theorem, we have:
$$a^2+c^2= (\frac{11}{2})^2 + (\frac{5}{2})^2 = \frac{125}{4}$$
Since $a$ is positive, we take the positive square root to get $a=\frac{\sqrt{125}}{2}=\frac{5\sqrt{5}}{2}.$ Thus, the length of the major axis is $2a=2\cdot\frac{5\sqrt{5}}{2}=5\sqrt{5}.$
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To find the length of the major axis of the ellipse, we need to determine the distance between the points where the ellipse intersects the $x$-axis.
Let's denote the coordinates of the center of the ellipse as $(h, k)$. Since the ellipse is tangent to the $x$-axis, the distance between the center and the $x$-axis is equal to the distance between the center and one of the points of tangency.
By symmetry, we can see that the center of the ellipse lies at $(2, -5)$, which is exactly halfway between the two foci.
Now, to find the distance between the center of the ellipse and the $x$-axis, we need to calculate the difference between the $y$-coordinate of the center and the $y$-coordinate of either of the foci. In this case, we can see that it is $|-5 - (-3)| = 2$. This is the length of the minor axis of the ellipse.
The major axis of an ellipse is the longest diameter, which is determined by the distance between the points where the ellipse intersects the $x$-axis. Since we already found the length of the minor axis, we just need to multiply it by 2 to find the length of the major axis.
Therefore, the length of the major axis is $2 \times 2 = \boxed{4}$.