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# A certain ellipse is tangent to both the -axis and the -axis, and its foci are at and ​ Find the length of the major axis.

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A certain ellipse is tangent to both the $$x$$-axis and the $$y$$-axis, and its foci are at $$(2, -3 + \sqrt{5})$$ and $$(2, -3 - \sqrt{5})$$ Find the length of the major axis.

May 7, 2019

#1
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The major axis will lie parallel to the y axis

So we have this form   :  (y - k)^2   +  (x - h)^2

_______       _______   =    1

a^2                b^2

The center of the ellipse  is  (2, - 3)  = (h, k)

Since the ellipse is tangent to the y axis and (2, -3) is the center.....then the minor axis must be 4 units in length

So  b = 4/2  = 2   and b^2  = 2

And we can find "a" thusly :   a^2 - b^2  = c^2

a^2  = ???

b^2 = 4

c^2 = (√5)^2  = 5

So

a^2 - 4 = 5

a^2  = 9

a = 3

And the major axis  = 2a  = 2(3)  = 6

Here's a graph : https://www.desmos.com/calculator/iagepa6zop   May 7, 2019
#3
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Thanks only thing what "thusly" is that a math term I should know or is it just like a therefore. I know it is a dumb question I should know the answer to.

May 13, 2019