Anika W. answered • 04/14/21
Patient and Caring Physics Major with 6+ Years Tutoring Experience
Hi Polina!
Since the y values of the vertices aren't changing, only the x values, it means it's a horizontal ellipse.
Equation for horizontal major axis ellipse:
(x-h)2/a2 + (y-k)2/b2 = 1
The center coordinate is (h,k) so we can do
|-2 - 10 |/2 = |-12|/2 =12/2 = 6 (this will also be useful as our "a" in our equation)
This is the distance between the 2 x coordinates divided by 2.
The vertex is +6 away from -2 and -6 away from 10.
The y coordinate stays the same.
(h,k) = (-2+6,-4) = (4,-4)
So far we have (x-4)2/a2 + (y+4)2/b2 = 1
a and b are the distance from the center to the vertex and co-vertex, respectively. We aren't given b though so we will have to use c, which is the distance from the center to the focus.
c2 = a2 - b2
center to vertex is 6, so a = 6
center to focus is 3. ( (4,-4) to (7,-4) is a difference of 3) so c = 3
32 = 62 - b2
9 = 36 - b2
b = sqrt(27)
plugging it all back into the equation:
(x-4)2/62 + (y+4)2/(sqrt(27))2 = 1
(x-4)2/36 + (y+4)2/27 = 1
