Please Help! Write the equation for the ellipses described below:

Question

Write the equation for an ellipse with vertices at (2,-5), (2,3); Foci: (2,-4), (2,2)

Write the equation for another ellipse with Major axis vertices: (2,2), (−4,2); Minor axis vertices: (−1,1), (−1,3)

Answer 1

2c = 2 - (-4) = 6; c = 3; 2a = 3 - (-5) = 8; a = 4; b² = 16 - 9 = 7; centerat (2, -1)

Equation: (x - 2)²/7 + (y + 1)²/16 = 1

Answer 2

Hi Surah k

For your other Ellipse, you are given the coordinates along the Major Axes (2,2),(-4,2) and the coordinates of the Minor Axes (-1,1), (-1,3)

Use the midpoint formula to get the coordinates for the center of your ellipse

By the way both sets of vertices should give the same midpoint

Once you have those coordinates you also have (h,k)

Next check the rules for ellipses in general against your information to see if your Major Axes is parallel to the x or the y axes. Once this is known apply the relationships between h, k, a, and b for your parallel axis.

You can also use the distance formula to calculate lengths of the Major and Minor axes and solve for a and b

Remember to graph your ellipse to confirm the vertices and the center

Center (-1,2)

Ellipse (x+1)²/9 + (y-2)²/1 = 1

https://www.desmos.com/calculator/wrjsevb9fc

Above is a link to the graph confirming your vertices and the center

Hope this helps, give it a try to see if you get the same.