Identify the type of conic section and coordinates of the center, foci and vertices.

Question

(y+3)^2/25+(x-2)^2/16=1

Answer 1

Since both quadratic terms are positive, it is an ellipse.    The center is at (2,-3).   The major axis is parallel to the y axis with the equation x = 2.   Thus the x coordinates of the vertices and foci are all equal to 2.

 

The length of the semi-major axis, a, is 5.  The length of the semi-minor axis, b, is 4.   c = sqrt(a² -b²) = 3

 

The distance from the center to a vertex is a = 5,   Thus the y coordinates of the vertices are  -3 ± 5.  Therefore the coordinates of the vertices are (2, -8) and (2,2).

 

The distance from the center to a focal point is c = 3.  Thus the y coordinates of the foci are -3 ± 3.  Therefore the coordinates of the foci are  (2, -6) and (2,0).