Question

Given the foci and vertices of an ellipse, complete the following steps.

Vertices at (4, -6 +or- 9)
Foci at (4,6 +or- 5√2)
what is the center of the ellipse? Explain how to determine this from the given points.

Answer 1

The center of the ellipse is at (4, -6). It is calculated by finding the midpoint of the y-coordinates of the vertices while the x-coordinate stays the same since the vertices share the same x-coordinate.

Step-by-step explanation:

The student has been asked to find the center of an ellipse given the vertices and foci. The vertices are located at (4, -6 +or- 9) which means there are two vertices, one at (4, -6+9) and the other at (4, -6-9). The foci are at (4, 6 +or- 5√2), meaning we have two foci, one at (4, 6+5√2) and the other at (4, 6-5√2).

The center of an ellipse is the midpoint between its vertices and lies along the line segment that connects its foci. By examining the coordinates of the vertices, we can see that they share the same x-coordinate, so the x-coordinate of the center will also be the same, which is 4. For the y-coordinate, we take the average of the y-coordinates of the vertices: (-6 + 9 + (-6 - 9))/2 which simplifies to -6. Therefore, the center of the ellipse is at (4, -6).