Ask question

# Foci: (0,0), (0,8), major axis of length 16 # Foci: (0,0), (0,8), major axis of length 16

Question
Solid Geometry asked 2020-10-28
Foci: (0,0), (0,8), major axis of length 16

## Answers (1) 2020-10-29
The foci are in the vertical direction so the ellipse is vertical. Its equation must then be of the form $$\displaystyle{\left(\frac{{\left({x}−{h}\right)}^{{2}}}{{b}^{{2}}}\right)}+{\left(\frac{{\left({y}−{k}\right)}^{{2}}}{{a}^{{2}}}\right)}={1}.$$
The center of the ellipse is the midpoint between the foci. Since the foci are (0,0) and (0,8), the center is (h,k)=((0+0)/2,(0+8)/2)=(0/2,8/2)=(0,4). The distance from (0,4) to (0,0) is 4 so c=4
The length of the major axis of an ellipse is 2a so the length of the major axis is 16, then 2a=16. Dividing both sides by 2 then gives a=8.
The values of aa, bb, and cc are related by the equation c2=a2−b2. Substitute the values of a=8 and c=4 into this equation to find b2:
$$\displaystyle{c}^{{2}}={a}^{{2}}-{b}^{{2}}$$
$$\displaystyle{4}^{{2}}={8}^{{2}}-{b}^{{2}}$$
$$\displaystyle{16}={64}-{b}^{{2}}$$
$$\displaystyle-{48}=-{b}^{{2}}$$
$$\displaystyle{48}={b}^{{2}}$$
Substituting in h=0, k=4, a=8 and b2=48 then gives:
$$\displaystyle{\left(\frac{{\left({x}-{h}\right)}^{{2}}}{{b}^{{2}}}\right)}+\frac{{{\left({y}-{k}\right)}^{{2}}}}{{a}^{{2}}}={1}$$
$$\displaystyle{\left(\frac{{\left({x}-{0}\right)}^{{2}}}{{48}}\right)}+\frac{{{\left({y}-{4}\right)}^{{2}}}}{{8}^{{2}}}={1}$$
$$\displaystyle{\left({x}^{{2}}+{48}\right)}+\frac{{{\left({y}-{4}\right)}^{{2}}}}{{64}}={1}$$

### Relevant Questions asked 2021-01-10
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3,3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2,4) and focus at (-4, 4) asked 2021-01-06
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3, 3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2, 4) and focus at (-4, 4) asked 2021-02-21
Find the area of the regular polygon with the given apothem a and side length s. decagon, a = 26 m, s = 16.9 m asked 2021-01-06
The gazebo in the photo is built in the shape of a regular octagon. Each side is 6 ft long, and the enclosed area is 172.8 ft². What is the length of the apothem? asked 2021-02-03
Rectangle CDEF ~ rectangle GHJK, and the simularity ratio of CDEF to GHJK is $$\displaystyle\frac{{1}}{{16}}$$. If DE=20 what is HK? asked 2020-12-15
The area of a rectangular cloth is (6x2 - 19x - 85) cm2. The length is (2x + 5) cm. Find the width. asked 2020-12-16
The gazebo in the photo is built in the shape of a regular octagon. Each side is 13ft long and the enclosed area is 816.4ft^2
What is the length of the apothem? asked 2020-11-02
The perimeter of a rectangle is 56 in. The ratio of the length to the width is 6:1. Find the length and the width. asked 2021-01-28
Find the area of the regular polygon with the given apothem a and side length s. decagon, a = 10 m, s = 6.5 m asked 2020-12-17
Salim built a rectangualr prism with a length of 5 inches, a width of 4 inches, and a height of 3 inches. Would the prism Natalie built with a length of 3 inches, a width of 4 inches, and a height of 5 inches have the same volume or a different volume than Salim's prism? Explain
...