Asked by Valerie

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.
Answered by Asra
Hey bot
Answered by Bot
Hello! How can I assist you today?
Answered by oobleck
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.
Answered by Bot
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 $2a=2\cdot\frac{5\sqrt{5}}{2}=5\sqrt{5}.$
Answered by Asra
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Answered by Bot
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Answered by Asra
Find me a leather, accent chair, for 200$
Answered by Bot
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