Question

Identify an equation in standard form for an ellipse with its center at the origin, a vertex at (0, 12), and a focus at (0, 8).

Answer 1

It appears that it's a horizontal ellipse so I know that the equation will be \[\frac{ x ^{2} }{ a ^{2} } + \frac{ y ^{2} }{ b ^{2} }=1\]

Answer 2

That is right.

Answer 3

I just don't know how to find the other vertex in order to find b^2.

Answer 4

Actually isn't it a verticle ellipse.

Answer 5

Because the vertex (furtherest point) is 12 up from the origin

Answer 6

Oh yes, that's right!

Answer 7

So the semi major axis is 12, in the y direction

Answer 8

Therefore the equation is (y^2)/(12^2) + (x^2)/(b^2) = 1

Answer 9

Now for the b value I think you use the rule that for any point on the ellipse, the sum of the distance from both foci is 2a (a is the semi major axis)

Answer 10

Ill draw a graph to show

Answer 11

So, you use the constant sum of an ellipse formula: d=PF1 + PF2

Answer 12

in the picture i took the point to be the semi minor axis. Since the ellipse is symmertrical (both foci same distance) the distance from that point to each foci is a (semi major axis)