find equation of the ellipse whose center is at the origin and foci f(2,2) f(-2,-2) and length of semi-major axis is square root of 8?

The ellipse is degenerate, since the foci are at the vertices.

student

do it for the first one

To find the equation of an ellipse with the given information, we can follow these steps:

Step 1: Remember that the standard equation of an ellipse with center (h, k), semi-major axis length 'a', and semi-minor axis length 'b' is given by:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

Step 2: Since the center is given as the origin (0, 0), the equation simplifies to:
x^2 / a^2 + y^2 / b^2 = 1

Step 3: We know that the distance between the center of the ellipse and each focus equals the distance between each focus and a point on the ellipse. Using this information, we can find the value of 'a'.

Step 4: The distance between the center and any focus on an ellipse is given by the formula:
c = sqrt(a^2 - b^2)

Step 5: In this case, the foci are given as f(2, 2) and f(-2, -2).

- For the first focus, f(2, 2):
c^2 = (2 - 0)^2 + (2 - 0)^2
= 4 + 4
= 8

- For the second focus, f(-2, -2):
c^2 = (-2 - 0)^2 + (-2 - 0)^2
= 4 + 4
= 8

Step 6: Since both foci have the same value of 'c^2', we can equate them:
8 = a^2 - b^2

Step 7: The length of the semi-major axis, a, is given as the square root of 8:
a = √8

Step 8: Substituting the value of 'a' into the previous equation:
8 = (√8)^2 - b^2
8 = 8 - b^2
b^2 = 0

Step 9: Since b^2 equals zero, the ellipse degenerates into a circle, which means a = b = √8.

Step 10: Finally, substituting the values of a and b into the equation from Step 2, we have:
x^2 / (√8)^2 + y^2 / (√8)^2 = 1

Simplifying further, we have:
x^2 / 8 + y^2 / 8 = 1

Therefore, the equation of the ellipse is:
x^2 + y^2 = 8