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1) Find the center, foci, vertices, and eccentricity of the of the ellipse
and sketch graph of the follwing equation: .$\displaystyle 16x^2 + 25y^2 - 64x + 150y + 279 \:=\:0$
Solution
$\displaystyle \begin{array}{cc}\text{Center:} & (2,\,-3) \\
\text{Vertices:} & (2 \pm0.791,\,-3) \\
\text{Foci:} & (2 \pm 0.474,\,-3) \\
\text{Eccentricity: } & \frac{0.474}{0.791}\end{array}$
2) Find the equation of each ellipse given the following:
$\displaystyle a)\;\begin{array}{cc}\text{Vertices:} & (0,\, 2),\;(4,\,2)\\
\text{Eccentricity:} & \frac{1}{2}\end{array}$
Solution:. . $\displaystyle \frac{(x - 2)^2}{4} + \frac{(y - 2)^2}{3} \;=\;1$
$\displaystyle b)\;\begin{array}{cc}\text{Foci:} & (0,\,\pm5)\\
\text{Major axis:} & 14\end{array}$
Solution: .$\displaystyle \frac{x^2}{24} + \frac{y^2}{49} \;=\;1$
$\displaystyle c)\;\begin{array}{cc}\text{Center:} & (1,\,2) \\
\text{Major axis:} & \text{vertical} \\
\text{points on ellipse:} & (1,\,6),\;(3,\,2) \end{array}$
Solution: .$\displaystyle \frac{(x -1)^2}{4} + \frac{(y -2)^2}{16} \;=\;1$