Find the vertices and foci of the ellipse and sketch its graph.?

Question

Find the vertices and foci of the ellipse and sketch its graph:
x^2 + 2y^2 - 6x + 4y + 7 = 0

Answer 1

x^2 + 2y^2 - 6x + 4y + 7 = 0
Complete the square on x and y:
x^2 - 6x + 9 + 2y^2 + 4y + 2 = - 7 + 9 + 2
(x - 3)^2 + 2(y + 1)^2 = 4
(x - 3)^2/4 + (y + 1)^2/2 = 1
Horizontal major axis, centered at (3, -1)
a^2 = 4
a = 2
b^2 = 2
b = √2
a^2 = b^2 + c^2
c = √(a^2 - b^2) = √(4 - 2) = √2
Vertices at 3 +/- 2 ===> 1, 5
(1, -1), (5, -1)
Foci at:
(3 +/- √2, -1)
Graph left to the student.