How to get the foci and vertices of an ellipse with the equation of (x+3)^2/24 + (y-5)^2/49=1?

I'm sure if you review the properties of ellipses you can come up with the values given at

http://www.wolframalpha.com/input/?i=ellipse+(x%2B3)%5E2%2F24+%2B+(y-5)%5E2%2F49%3D1

To find the foci and vertices of an ellipse given its equation, follow these steps:

Step 1: Examine the equation to identify the major and minor axes.
- In the given equation, (x+3)^2/24 + (y-5)^2/49 = 1, the denominator of the x-term is larger than the denominator of the y-term. Therefore, the major axis is along the x-axis, and the minor axis is along the y-axis.

Step 2: Find the length of the major and minor axes.
- The length of the major axis 2a is equal to the square root of the denominator of the x-term multiplied by 2. In this case, 2a = 2 * sqrt(24) ≈ 9.798.
- The length of the minor axis 2b is equal to the square root of the denominator of the y-term multiplied by 2. In this case, 2b = 2 * sqrt(49) = 14.

Step 3: Determine the center of the ellipse.
- The center of the ellipse is given by the coordinates opposite the signs in the equation. In this case, the coordinates of the center are (-3, 5).

Step 4: Calculate the values of a² and b².
- In the equation, (x+3)^2/24 + (y-5)^2/49 = 1, a² is equal to the denominator of the x-term (24), and b² is equal to the denominator of the y-term (49).

Step 5: Identify the coordinates of the foci.
- The coordinates of the foci are given by (h±c, k), where (h, k) are the coordinates of the center and c is the distance from the center to the foci. The value of c can be calculated using the equation c = sqrt(a² - b²).
- Substitute the values of a² = 24 and b² = 49 into the equation to calculate c. In this case, c = sqrt(24 - 49) = sqrt(-25).
- The imaginary value for c suggests that the given ellipse does not have real foci. Therefore, there are no foci for this ellipse.

Step 6: Determine the coordinates of the vertices.
- The coordinates of the vertices can be found by adding and subtracting a from the x-coordinate of the center, and the same with b and the y-coordinate of the center. In this case, the coordinates of the vertices are:
- Vertex 1: (-3 + a, 5) ≈ (6.798, 5)
- Vertex 2: (-3 - a, 5) ≈ (-12.798, 5)

Hence, the foci for the given ellipse do not exist, and the coordinates of the vertices are approximately (6.798, 5) and (-12.798, 5).