Question

how do you create an equation of an ellipse with a major axis of length 18 and foci located at (4,7) and (4,11)

Answer 1

do you know what this looks like? we need that first

Answer 2

an ellipse, but how can you tell if its horizontal or vertical?

Answer 3

because of the foci
the \(x\) values are the same, whereas the \(y\) values are different

Answer 4

so it is going to look like
\[\frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1\] you need the center, which is half way between \((4,7)\) and \((4,11)\) and also \(a\) and \(b\)

Answer 5

you got the center?

Answer 6

(4,9)

Answer 7

k good
so now we are here
\[\frac{(y-9)^2}{a^2}+\frac{(x-4)^2}{b^2}=1\]

Answer 8

how far from the center to the top part of the ellipse?

Answer 9

9

Answer 10

k good, so now we know \(a=9\) and so \(a^2=81\)

Answer 11

now we are here
\[\frac{(y-9)^2}{9^2}+\frac{(x-4)^2}{b^2}=1\]

Answer 12

how far from the center to the foci?

Answer 13

2

Answer 14

b^2=4?

Answer 15

got it
so \(c=2\) and therefore \(c^2=4\)
you need \(b^2=a^2-c^2\)

Answer 16

no that was \(c\) not \(b\)
we still have to find \(b\)
or actually \(b^2\)

Answer 17

b^2 = the square root of 77?

Answer 18

i think actually \(b^2=77\) not \(b=77\)

Answer 19

I get it! thank you for all the help now I understand it.

Answer 20

lets check
looks good
http://www.wolframalpha.com/input/?i=ellipse+%28y-9%29^2%2F81%2B%28x-4%29^2%2F77%3D1 \yw