ellipse with foci at (-4, 0) and (-4, -6) and major axis of 10?

Question

Write the equation of an ellipse with foci at (-4, 0) and (-4, -6) and major axis of 10

Answer 1

Hi,
Well, the major axis is on x=-4, so the equation will be of the following form:
(y-k)²/a² +(x-h)²/b² = 1
So, a = ½ the major axis:
a =( ½)10
=5
Now, let’s find b from the this equation:
b² =a² -c²….(Where the focus is the distance from the focus to the center.)
c=(1/2)0-6 =
=-3
So,
b²= 5² -(-3)²
=25-9
=16
So, we now have this:
(y-k)²/25 +(x-h)²/16 = 1
Now, obviously h = -4 from the foci locations, and k = -3 taking ½ the distance between the foci and subtracting that from zero, the upper focus.
So, we finally have this:
(y+4)²/25 +(x+3)²/16 = 1

Hope this helps.
FE