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**Unformatted text preview: **moved farther in the x
direction than in the y direction, the major axis will lie along the horizontal line y = −3, which
means the minor axis lies along the vertical line x = 1. The vertices are the points on the ellipse
which lie along the major axis so in this case, they are the points (−1, −3) and (3, −3). To ﬁnd
√
√
√
the foci, we ﬁnd c = 4 − 1 = 3, which means the foci lie 3 units from the center. Since the
√
√
major axis is horizontal, the foci lie 3 units to the left and right of the center, at (1 − 3, −3)
√
and (1 + 3, −3). Plotting all of this information gives
y 1 2 3 4 x −1
−2
−3
−4 As you come across ellipses in the homework exercises and in the wild, you’ll notice they come in
all shapes in sizes. Compare the two ellipses below. Certainly, one ellipse is ‘rounder than the other. This notion of roundness is quantiﬁed below.
Definition 7.5. The eccentricity of an ellipse, denoted e, is the following ratio:
e= distance from the center to a focus
distance from...

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