minor axis 12, distance between foci 16
you have
b=6
c=8
so, a=10 and e=4/5
To find the major axis and the coordinates of the foci of an ellipse given the minor axis and the distance between the foci, you can use the following steps:
Step 1: Identify the minor axis: The minor axis is the axis that is perpendicular to the major axis and passes through the center of the ellipse. In this case, the minor axis is given as 12.
Step 2: Calculate the value of b: The minor axis, which is 2b, is given as 12. Therefore, b = 12/2 = 6.
Step 3: Calculate the value of c: The distance between the foci is given as 16. Since the distance between the center and each focus is equal to c, c = 16/2 = 8.
Step 4: Calculate the value of a: In an ellipse, a is half of the major axis. Since c and b are the distances from the center to the foci and the vertices respectively, a can be found using the formula a^2 = b^2 + c^2. This gives a^2 = 6^2 + 8^2 = 36 + 64 = 100. Taking the square root of both sides, a = 10.
Step 5: Determine the major axis: The major axis is equal to 2a. Therefore, the major axis is 2 * 10 = 20.
Step 6: Determine the coordinates of the foci: The foci are located a distance of c units from the center along the major axis. Since the center of the ellipse is at the origin (0, 0), the coordinates of the foci are (+/- c, 0). In this case, the coordinates of the foci are (+/- 8, 0).
In summary,
- The major axis is 20.
- The minor axis is 12.
- The coordinates of the foci are (+/- 8, 0).
To find the major axis of an ellipse given the length of the minor axis and the distance between the foci, you can use the formula:
major axis = 2 * √(minor axis^2 + (distance between foci)^2)
In this case, you are given:
minor axis = 12
distance between foci = 16
Substituting these values into the formula, we have:
major axis = 2 * √(12^2 + 16^2)
= 2 * √(144 + 256)
= 2 * √400
= 2 * 20
= 40
Therefore, the major axis of the ellipse is 40.