Answer:
Step-by-step explanation:
1. Equation of an ellipse is:
(x - h)² / a² + (y - k)² / b² = 1
where (h, k) is the center and a and b are the length of half the minor/major axes.
The center is the midpoint of the foci:
(h, k) = (½ (2+8), ½(0+0))
(h, k) = (5, 0)
The foci have the same y-coordinate, so the horizontal axis is the major axis:
a = 12/2
a = 6
The distance from the foci to the center is c:
c = 8-5
c = 3
b can be found using the formula:
c² = a² - b²
3² = 6² - b²
b² = 36 - 9
b² = 27
So the equation is:
(x - 5)² / 36 + (y - 0)² / 27 = 1
2. Same steps as #1. First find the center:
(h, k) = (½ (1+5), ½ (2+2))
(h, k) = (3, 2)
The foci have the same y-coordinate, so the horizontal axis is the major axis:
a = 6/2
a = 3
The distance from the foci to the center is c:
c = 5-3
c = 2
b can be found using the formula:
c² = a² - b²
2² = 3² - b²
b² = 9 - 4
b² = 5
So the equation is:
(x - 3)² / 9 + (y - 2)² / 5 = 1
3. The vertices have the same y coordinate, so this is a horizontal hyperbola:
(x - h)² / a² - (y - k)² / b² = 1
The center (h, k) is the midpoint of the vertices:
(h, k) = (½ (-2+2), ½ (0+0))
(h, k) = (0, 0)
The distance from the center to the vertices is a:
a = 2-0
a = 2
The distance from the center to the foci is c:
c = 6-0
c = 6
b can be found using the formula:
c² = a² + b²
6² = 2² + b²
b² = 36 - 4
b² = 32
So the equation is:
(x - 0)² / 4 - (y - 0)² / 32 = 1
