Write an equation for the ellipse that satisfies the set of conditions.
endpoint of major axis at ( 0, 10) and (0, -10) foci at (0, 8) and (0-8)
Please help, I really don't understand how to do this
an ellipse with center at (h,k) and semi-axes a and b is
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
the foci are at c where a^2 = b^2 + c^2 and lie on the major axis.
you thus have
a = 10
c = 8
so, b = 6
It appears that the center is at (0,0), so
x^2/100 + y^2/36 = 1
To write the equation for an ellipse that satisfies a given set of conditions, you need to understand the properties of an ellipse and its standard equation.
An ellipse is a closed curve consisting of all points such that the sum of the distances from any point on the ellipse to two fixed points (called foci) is constant. The longest distance across the ellipse is called the major axis, and the shortest distance across the ellipse is called the minor axis.
The standard equation for an ellipse with center (h, k), horizontal major axis, and length 2a is given by:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
To find the values of a, b, h, and k for the given conditions, follow these steps:
1. The endpoints of the major axis are (0, 10) and (0, -10). Since the ellipse is vertically oriented, the major axis runs vertically. Therefore, the length of the major axis is 2a = 10 - (-10) = 20. Hence, a = 20 / 2 = 10.
2. The foci are located at (0, 8) and (0, -8). Note that these points are on the major axis, a distance of c from the center. To find c, use the relationship c^2 = a^2 - b^2, where b is the distance from the center to the vertices along the minor axis. Since the center is at (0, 0), c^2 = 10^2 - b^2.
3. Given that the foci are at (0, 8) and (0, -8), we see that c = 8. Substituting this into the equation c^2 = a^2 - b^2, we get 8^2 = 10^2 - b^2 => 64 = 100 - b^2 => b^2 = 100 - 64 = 36.
4. Now we have all the necessary values. Since the center is at (0, 0) and a = 10, b = 6, the equation of the ellipse is:
(x - 0)^2 / 10^2 + (y - 0)^2 / 6^2 = 1
Simplifying further, we have:
x^2 / 100 + y^2 / 36 = 1
Therefore, the equation of the ellipse that satisfies the given conditions is x^2 / 100 + y^2 / 36 = 1.