Geometric Definition of a Parabola

A parabola is the set of points in the plane that are equidistant from a fixed point \(F\) (the focus) and a fixed line \(l\) (the directrix).

Graphs of Parabolas with Vertex at the Origin

A parabola with vertex at the origin has one of the following standard equations.

Vertical Axis

\[ x^{2}=4 p y \]

Focus \((0, p)\), directrix \(y=-p\)

Horizontal Axis

\(y^{2}=4 p x\)

Focus \((p, 0)\), directrix \(x=-p\)

Geometric Definition of an Ellipse

An ellipse is the set of all points in the plane for which the sum of the distances to each of two given points \(F_{1}\) and \(F_{2}\) (the foci) is a fixed constant.

Graphs of Ellipses with Center at the Origin

An ellipse with center at the origin has one of the following standard equations.

Horizontal Axis

\[ \begin{aligned} & \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \\ & (a>b>0) \end{aligned} \]

Foci \(( \pm c, 0), c^{2}=a^{2}-b^{2}\)

Eccentricity of an Ellipse

The eccentricity of an ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) or \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1(\) where \(a>b>0)\) is the number

\[ e=\frac{c}{a} \]

where \(c=\sqrt{a^{2}-b^{2}}\). The eccentricity \(e\) of any ellipse is a number between 0 and 1 . If \(e\) is close to 0 , then the ellipse is nearly circular; the closer \(e\) gets to 1 , the more elongated the ellipse becomes.

Geometric Definition of a Hyperbola

A hyperbola is the set of all points in the plane for which the absolute value of the difference of the distances to each of two given points \(F_{1}\) and \(F_{2}\) (the foci) is a fixed constant.

Graphs of Hyperbolas with Center at the Origin

A hyperbola with center at the origin has one of the following standard equations.

Horizontal Axis

\[ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \]

Vertical Axis

\[ -\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1 \]

Foci \(( \pm c, 0), c^{2}=a^{2}+b^{2}\)

\[ \text { Asymptotes: } y= \pm \frac{b}{a} x \]

\(\operatorname{Foci}(0, \pm c), c^{2}=a^{2}+b^{2}\)

\[ \text { Asymptotes: } y= \pm \frac{a}{b} x \]

Shifted Conics

If the vertex of a parabola or the center of an ellipse or a hyperbola does not lie at the origin but rather at the point \((h, k)\), then we refer to the curve as a shifted conic. To find the equation of the shifted conic, we use the "unshifted" form for the appropriate curve and replace \(x\) by \(x-h\) and \(y\) by \(y-k\).

General Equation of a Shifted Conic

The graph of the equation

\[ A x^{2}+C y^{2}+D x+E y+F=0 \]

(where \(A\) and \(C\) are not both 0 ) is either a conic or a degenerate conic. In the nondegenerate cases the graph is

  1. a parabola if \(A=0\) or \(C=0\),
  2. an ellipse if \(A\) and \(C\) have the same sign (or a circle if \(A=C\) ),
  3. a hyperbola if \(A\) and \(C\) have opposite signs.

To graph a conic whose equation is given in general form, complete the squares in \(x\) and \(y\) to put the equation in standard form for a parabola, an ellipse, or a hyperbola.

Rotation of Axes

Suppose the \(x\) - and \(y\)-axes in a coordinate plane are rotated through the acute angle \(\phi\) to produce the \(X\) - and \(Y\)-axes, as shown in the figure. Then the coordinates of a point in the \(x y\) - and the \(X Y\)-planes are related as follows:

\[ \begin{array}{ll} x=X \cos \phi-Y \sin \phi & X=x \cos \phi+y \sin \phi \\ y=X \sin \phi+Y \cos \phi & Y=-x \sin \phi+y \cos \phi \end{array} \]

The General Conic Equation

The general equation of a conic is of the form

\[ A x^{2}+B x y+C y^{2}+D x+E y+F=0 \]

The quantity \(B^{2}-4 A C\) is called the discriminant of the equation. The graph is

  1. a parabola if \(B^{2}-4 A C=0\),
  2. an ellipse if \(B^{2}-4 A C<0\),
  3. a hyperbola if \(B^{2}-4 A C>0\).

To eliminate the \(x y\)-term in the general equation of a conic, rotate the axes through an angle \(\phi\) that satisfies

\[ \cot 2 \phi=\frac{A-C}{B} \]

Polar Equations of Conics

A polar equation of the form

\[ r=\frac{e d}{1 \pm e \cos \theta} \quad \text { or } \quad r=\frac{e d}{1 \pm e \sin \theta} \]

represents a conic with one focus at the origin and with eccentricity \(e\). The conic is

  1. a parabola if \(e=1\),
  2. an ellipse if \(0<e<1\),
  3. a hyperbola if \(e>1\).