The Ellipse

The Ellipse

An ellipse is the locus of a point moving such that the sum of its distances from two fixed points (called foci) is constant. This constant sum is equal to the length of the major axis ($2a$).

Key Components

• Center ($C$): The midpoint of the foci.
• Vertices ($V$): The endpoints of the major axis.
• Co-vertices ($B$): The endpoints of the minor axis.
• Major Axis: The longer axis ($2a$).
• Minor Axis: The shorter axis ($2b$).
• Foci ($F$): Two fixed points on the major axis.
• Focal Length ($c$): The distance from the center to a focus.
• Relationship: $a^2 = b^2 + c^2$ (where $a > b$).

Standard Equations

Let $(h, k)$ be the center.

Horizontal Ellipse (Major Axis is Horizontal)

Horizontal Ellipse Equation

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• $a > b$
• Center: $(h, k)$
• Vertices: $(h \pm a, k)$
• Co-vertices: $(h, k \pm b)$
• Foci: $(h \pm c, k)$ where $c = \sqrt{a^2 - b^2}$

Vertical Ellipse (Major Axis is Vertical)

Vertical Ellipse Equation

$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

• $a > b$
• Center: $(h, k)$
• Vertices: $(h, k \pm a)$
• Co-vertices: $(h \pm b, k)$
• Foci: $(h, k \pm c)$ where $c = \sqrt{a^2 - b^2}$

Eccentricity

The eccentricity $e$ of an ellipse measures its deviation from being circular. For an ellipse, $0 < e < 1$.

Eccentricity Formula

$e = \frac{c}{a}$

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Solved Problems