what is the equation of the ellipse with goci at (0,-4) and (0,4) and the sum of its focal radii being 10?
The major axis will be along the y-axis
with c = 4, 2a = 10
a = 5
by Pythagoras for this ellipse, b = 5
so ...
x^2/9 + y^2/25 = 1
Goci=foci, the plural of focus.
To find the equation of the ellipse, we can start by determining the center, major axis, and minor axis of the ellipse using the given information.
The center of the ellipse is the midpoint between the two foci, which in this case is (0, 0) since the foci are located at (0, -4) and (0, 4).
The major axis of the ellipse is the line segment that connects the two foci. In this case, the major axis is vertical since the two foci have the same x-coordinate (0).
The length of the major axis is twice the distance between the center and either of the foci. In this case, the distance between the center (0, 0) and either focus (0, -4) or (0, 4) is 4. Therefore, the length of the major axis is 2 * 4 = 8.
The minor axis of the ellipse is the line segment perpendicular to the major axis and passing through the center. In this case, the minor axis is horizontal since the major axis is vertical.
Now that we have the center, major axis, and minor axis, we can write the equation of the ellipse.
The standard equation for an ellipse centered at the origin (h, k) with a vertical major axis of length 2a and a horizontal minor axis of length 2b is:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
In this case, the center is (0, 0), the major axis length is 8 (2a), and the minor axis length is 2b (since the length of the minor axis is usually smaller).
Substituting these values into the equation, we have:
(x-0)^2/4^2 + (y-0)^2/b^2 = 1
x^2/16 + y^2/b^2 = 1
Now, we need to find the value of b to complete the equation. The sum of the focal radii is given as 10. The focal radii are the distances from any point on the ellipse to both foci. Since the foci are located at (0, -4) and (0, 4), the sum of the focal radii can be expressed as:
2a = 10
a = 10/2 = 5
We know that a^2 = b^2 + c^2, where c is the distance from the center to each focus. In this case, c = 4 (the distance between the center and either focus). We can solve for b as follows:
5^2 = b^2 + 4^2
25 = b^2 + 16
b^2 = 25 - 16
b^2 = 9
b = √9
b = 3
Now we have all the values we need to finalize the equation:
x^2/16 + y^2/9 = 1
Therefore, the equation of the ellipse with foci at (0, -4) and (0, 4) and the sum of its focal radii being 10 is x^2/16 + y^2/9 = 1.