Locate the foci with this equation:
(x - 5)² (y - 3)²
———————— + ————————— = 1
5² 10²
There are two forms of ellipses.
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1. Those that are longer horizontally and narrower vertically
These have the form:
(x - h)² (y - k)²
———————— + ———————— = 1
a² b²
a = semi-major axis, b = semi-minor axis, (h, k) = center
center = (h, k)
vertices = (h±a, k) _______
foci = (h±c, k) where c = Öa² - b²
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2. Those that are longer vertically and narrower horizontally
(x - h)² (y - k)²
———————— + ———————— = 1
b² a²
a = semi-major axis, b = semi-minor axis, (h, k) = center
center = (h, k)
vertices = (h, k±a) _______
foci = (h, k±c) where c = Öa² - b²
--------------------------------------------------------------
You can always tell which type ellipse you have because
the semi major axis a is always greater than the semi minor
axis b. Therefore a² will always be larger than b². If
a² is under the (x-h)², the ellipse is the first type. Otherwise
it is the second type.
Yours is the second type because the larger of 5² and 10² is
10² and it is underneath (y-k)².
a = 10
b = 5
center = (h, k) = (5. 3)
vertices = (h, k±a) = (5, 3±10), that is, (5, -7) and (5, 13)
To find the foci, we need to find c
_______
c = Öa² - b²
________
c = Ö10² - 5²
________
c = Ö100 - 25
__
c = Ö75
____
c = Ö25·3
_
c = 5Ö3
_
foci = (h, k±c) = (5, 3±5Ö3), that is
_ _
(5, 3-5Ö3) and (5, 3+5Ö3)
Your ellipse looks like this:
Edwin
AnlytcPhil@aol.com
