Write an equation in standard form for the ellipse with center (0,0), a Focus at (0,6), and a vertex at (0, -10).

Saturday, December 11, 2010

Write an equation in standard form for the ellipse with center (0,0), a Focus at (0,6), and a vertex at (0, -10).

We are given a focus at (0,6), the center at the origin, and a vertex at (0,-10), and we are asked for the standard form of the equation for the ellipse.

Let a be the distance from the center to a vertex on the major axis, b the distance from the center to a vertex on the minor axis, and c the distance from the center to one of the foci.

We know the major axis lies on the y-axis as the center is the origin and the foci lie on the y-axis.

Thus a=10. (The length of the major axis is 2a=20; the distance between the vertices on the major axis. By symmetry, the other vertex is at (0,10).)

Also c=6.

The relation between a,b, and c for an ellipse is $\displaystyle{{b}}^{{2}}={{a}}^{{2}}-{{b}}^{{2}},{a}>{b}$ so

$\displaystyle{{b}}^{{2}}={100}-{36}={64}=\Rightarrow{b}={8}$

Then the equation can be written as $\displaystyle\frac{{{x}}^{{2}}}{{64}}+\frac{{{y}}^{{2}}}{{100}}={1}$ or $\displaystyle\frac{{{y}}^{{2}}}{{100}}+\frac{{{x}}^{{2}}}{{64}}={1}$ depending on textbook/instructor preference.

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Saturday, December 11, 2010