Write equation of ellipse with foci (-1, 2) and (7, 2) and co-vertices (3, -1) and (3, 5) - show work

Answer 1

an ellipse is the locus of all points whose sum of distances to the foci is a constant

the verticies are the top and bottom of the ellipse on the vertical line x=3. Center of the ellipse is (3,2) midway between the two verticies. The foci are on a horizontal line y=2

any point on the ellipse, (x,y) is equal to the sum of distances to the foci, (-1,2) and (7,2)

that distance is 10. To get 10 from horizontal vertex to sum of distances to both foci, the horizontal co-verticies must be (-2,2) and (8,2)

(x-h)²/a² + (y-k)²/b² = 1, a=5, b=3 (h,k) is the center (3,2) a is the distance from the center to the left vertex and from the center to the right vertex b is the distance from the center to the top vertex and from the center to the bottom vertex

(x-3)²/25 + (y-2)²/9 = 1

or multiplying through by 25 times 9

9(x-3)² + 25(y-2)² = 225

If you want you can expand that further and combine constant terms to get

9x²-54x +25y² - 100y -44 = 0

but the first equation gives you the most information immediately about the vertices

The graph is a flattened ellipse, longer horizontally than vertically