Focus of a conic

Builds on Ellipse Hyperbola Parabola Circle

Definition

Let $Q$ be a non-degenerate conic section in the Euclidean plane $\mathbb{R}^2$ . Suppose that $Q$ is an ellipse or a hyperbola with the foci $F_1$ and $F_2$ , as in their definitions. Then $F_1$ and $F_2$ are the foci of $Q$ . If $Q$ is a circle, its centre is its only focus. If $Q$ is a parabola with the focus $F$ and a directrix, as in its definition, then $F$ is the only focus of $Q$ .

Remarks

The foci are determined by the conic. For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$ , the foci are $(\pm c, 0)$ with $c^2 = a^2 - b^2$ . For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ , the foci are $(\pm c, 0)$ with $c^2 = a^2 + b^2$ . Let $Q$ be a conic that is not a circle, and let $F$ be a focus of $Q$ . Then $Q$ is the set of the points whose distances to $F$ and to a line, the directrix, have a constant ratio $e > 0$ .

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Circle concept
Ellipse concept
Hyperbola concept
Parabola concept
Focus of a conic definition

Tag	`0RV3AEXPT6`
URI	`m2://0RV3AEXPT6`
Kind	definition
Writing	W0 stub
MSC	51N20
Defines	Focus of a conic
Uses	Ellipse, Hyperbola, Parabola, Circle

Focus of a conic

Definition

Remarks

Reading path

Further reading

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