Find the foci of the ellipse whose major axis has endpoints $(0,0)$ and $(13,0)$ and whose minor axis has length 12.
clearly the major axis has length 13, and the minor axis has length 12. So
a = 13/2
b = 6
c^2 = a^2-b^2 = 25/4, so c = 5/2
The center is at (13/2,0), so
h = 13/2 and k=0
The foci are at (h±c,k)
Since the major axis is horizontal, that means the equation is
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
(x - 13/2)^2/(13/2)^2 + y^2/6^2 = 1
see
http://www.wolframalpha.com/input/?i=ellipse+%28x+-+13%2F2%29^2%2F%2813%2F2%29^2+%2B+y^2%2F6^2+%3D+1
tysm
Find the foci of the ellipse whose major axis has endpoints $(0,0)$ and $(13,0)$ and whose minor axis has length 12.
the answers are (9,0) and (4,0).
To find the foci of an ellipse, we need to know the length of the major axis and the length of the minor axis. In this case, we are given that the major axis has endpoints $(0,0)$ and $(13,0)$, and the length of the minor axis is 12.
Step 1: Find the center of the ellipse
The center of an ellipse is the midpoint of the major axis. To find the midpoint, we can average the x-coordinates of the endpoints and the y-coordinates of the endpoints. In this case, the x-coordinates are 0 and 13, so the average is (0 + 13)/2 = 6.5. The y-coordinates are both 0, so the y-coordinate of the center is 0.
Therefore, the center of the ellipse is $(6.5, 0)$.
Step 2: Find the distance from the center to each focus
The distance from the center to each focus is given by the formula c = √(a^2 - b^2), where a is half the length of the major axis, and b is half the length of the minor axis.
In this case, the length of the major axis is 13, so a = 13/2 = 6.5. The length of the minor axis is 12, so b = 12/2 = 6.
Now we can calculate c: c = √(6.5^2 - 6^2) = √(42.25 - 36) = √6.25 = 2.5
Step 3: Find the foci
The foci of the ellipse are located at the points (h + c, k) and (h - c, k), where (h, k) is the center of the ellipse and c is the distance from the center to each focus.
In this case, the center of the ellipse is (6.5, 0) and the distance from the center to each focus is c = 2.5.
Therefore, the foci of the ellipse are $(6.5 + 2.5, 0)$ = $(9, 0)$ and $(6.5 - 2.5, 0)$ = $(4, 0)$.
So, the foci of the ellipse are $(9, 0)$ and $(4, 0)$.