Find an equation in standard form for the ellipse that satisfies the given conditions. Minor axis endpoints ​(0,±5​), major axis length 14

Question

Find an equation in standard form for the ellipse that satisfies the given conditions.

Minor axis endpoints

​(0,±5​),

major axis length 14

Answer 1

To find the standard form equation of an ellipse, you just need to know 3 things:

1. The center of the ellipse, which we call point (h,k)
2. The length of the horizontal radius, which we call a
3. The length of the vertical radius, which we call b

The standard form equation is:

(x-h)²/a² + (y-k)²/b² = 1 (remember, h is the x-value of the center point, k is the y-value of the center point, a is the horizontal radius of the ellipse [half the width of the ellipse], and b is the vertical radius of the ellipse [half the height of the ellipse])

So for this problem, we start by finding those 3 things we need:

1. The center of the ellipse is at (0,0) since the endpoints of one of the axes are at (0,5) and (0,-5)
2. Draw it out! Those points are both on the y-axis, and halfway between -5 and 5 is 0, so the center is at the origin!
3. The length of the horizontal radius must be 7, since the major axis (in this case horizontal, since we already know the minor axis is vertical) is 14. Half of 14 is 7, so the horizontal radius must be 7
4. The length of the vertical radius must be 5, as we already figured out from step 1

So, (h,k) is (0,0), a is 7, and b is 5, all of which we plug into our standard form equation:

(x-0)²/7² + (y-0)²/5² = 1

Now we simplify

x²/49 + y²/25 = 1

Answer 2

x^2/25 + y^2/49 = 1

or

49x^2 + 25y^2 = 1225