For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.
1. \(\frac{x^{2}}{25}+\frac{y^{2}}{64}=1\)
2. \(\frac{(x-2)^{2}}{100}+\frac{(y+3)^{2}}{36}=1\)
3. \(9 x^{2}+y^{2}+54 x-4 y+76=0\)
4. \(9 x^{2}+36 y^{2}-36 x+72 y+36=0\)
For the following exercises, graph the ellipse, noting center, vertices, and foci.
5. \(\frac{x^{2}}{36}+\frac{y^{2}}{9}=1\)
6. \(\frac{(x-4)^{2}}{25}+\frac{(y+3)^{2}}{49}=1\)
7. \(4 x^{2}+y^{2}+16 x+4 y-44=0\)
8. \(2 x^{2}+3 y^{2}-20 x+12 y+38=0\)
For the following exercises, use the given information to find the equation for the ellipse.
9. Center at (0,0) , focus at (3,0) , vertex at (-5,0)
10. Center at \((2,-2),\) vertex at \((7,-2),\) focus at (4,-2)
11. A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?