Find a Parabola with Vertex (3,1) Focus (4,1)
Find the vertex of 4y^2+4y-16x+13=0
Find the center of an ellipse 9x^2+4y^2-36x-24y-36=0
Find an ellipse with minor axis of 8 and vertices at (-9,3),(7,3)
In all of these problems complete the square, then look at standard form of conic section.
For example to find the center of the ellipse
Find the center of an ellipse 9x^2+4y^2-36x-24y-36=0
9 x^2 -36 x -36 = -4 y^2 + 24 y
(-9/4) x^2 + 9 x + 9 = y^2 -6 y
(-9/4) x^2 + 9 x + 18 = y^2 -6 y + 9
(-9/4) x^2 + 9 x + 18 = (y-3)^2
x^2 - 4 x - 8 = (-4/9) (y-3)^2
x^2 - 4 x = (-4/9) (y-3)^2 + 8
x^2 - 4 x + 2 = (-4/9) (y-3)^2 + 10
(x-2)^2 + (4/9) (y-3)^2 = 10
You can see right now that center is at (2,3)
(x-2)^2/10 + (y-3)^2/(90/4) = 1
x^2 - 4 x = (-4/9) (y-3)^2 + 8
x^2 - 4 x + 4 = (-4/9) (y-3)^2 + 12
(x-2)^2 + (4/9) (y-3)^2 = 12
You can see right now that center is at (2,3)
(x-2)^2/12 + (y-3)^2/(27) = 1