Essential Concepts

Understanding Polar Coordinates

• The polar coordinate system provides an alternative way to locate points in the plane.
• Convert points between rectangular and polar coordinates using the formulas
  
  [latex]x=r\cos\theta \text{ and }y=r\sin\theta[/latex]
  and
  
  [latex]r^2={x}^{2}+{y}^{2}\text{ and }\tan\theta =\frac{y}{x}[/latex].
• To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.
• Use the conversion formulas to convert equations between rectangular and polar coordinates.
• Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.

Area and Arc Length in Polar Coordinates

• The area of a region in polar coordinates defined by the equation [latex]r=f\left(\theta \right)[/latex] with [latex]\alpha \le \theta \le \beta[/latex] is given by the integral [latex]A=\frac{1}{2}{{\displaystyle\int }_{\alpha }^{\beta }\left[f\left(\theta \right)\right]}^{2}d\theta[/latex].
• To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.
• The arc length of a polar curve defined by the equation [latex]r=f\left(\theta \right)[/latex] with [latex]\alpha \le \theta \le \beta[/latex] is given by the integral [latex]L={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{\left[f\left(\theta \right)\right]}^{2}+{\left[{f}^{\prime }\left(\theta \right)\right]}^{2}}d\theta ={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{r}^{2}+{\left(\frac{dr}{d\theta }\right)}^{2}}d\theta[/latex].

Conic Sections

• The equation of a vertical parabola in standard form with given focus and directrix is [latex]y=\frac{1}{4p}{\left(x-h\right)}^{2}+k[/latex] where p is the distance from the vertex to the focus and [latex]\left(h,k\right)[/latex] are the coordinates of the vertex.
• The equation of a horizontal ellipse in standard form is [latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex] where the center has coordinates [latex]\left(h,k\right)[/latex], the major axis has length 2a, the minor axis has length 2b, and the coordinates of the foci are [latex]\left(h\pm c,k\right)[/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[/latex].
• The equation of a horizontal hyperbola in standard form is [latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex] where the center has coordinates [latex]\left(h,k\right)[/latex], the vertices are located at [latex]\left(h\pm a,k\right)[/latex], and the coordinates of the foci are [latex]\left(h\pm c,k\right)[/latex], where [latex]{c}^{2}={a}^{2}+{b}^{2}[/latex].
• The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.
• The polar equation of a conic section with eccentricity e is [latex]r=\frac{ep}{1\pm e\cos\theta }[/latex] or [latex]r=\frac{ep}{1\pm e\sin\theta }[/latex], where p represents the focal parameter.

Key Equations

• Area of a region bounded by a polar curve
  
  [latex]A=\frac{1}{2}{\displaystyle\int }_{\alpha }^{\beta }{\left[f\left(\theta \right)\right]}^{2}d\theta =\frac{1}{2}{\displaystyle\int }_{\alpha }^{\beta }{r}^{2}d\theta[/latex]
• Arc length of a polar curve
  
  [latex]L={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{\left[f\left(\theta \right)\right]}^{2}+{\left[{f}^{\prime }\left(\theta \right)\right]}^{2}}d\theta ={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{r}^{2}+{\left(\frac{dr}{d\theta }\right)}^{2}}d\theta[/latex]

Glossary

angular coordinate

[latex]\theta[/latex] the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (x) axis, measured counterclockwise

cardioid

a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is [latex]r=a\left(1+\sin\theta \right)[/latex] or [latex]r=a\left(1+\cos\theta \right)[/latex]

conic section

a conic section is any curve formed by the intersection of a plane with a cone of two nappes

directrix

a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two

discriminant

the value [latex]4AC-{B}^{2}[/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[/latex], is called a discriminant

eccentricity

the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix

focal parameter

the focal parameter is the distance from a focus of a conic section to the nearest directrix

focus

a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two

limaçon
the graph of the equation [latex]r=a+b\sin\theta[/latex] or [latex]r=a+b\cos\theta[/latex]. If [latex]a=b[/latex] then the graph is a cardioid

major axis

the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis

minor axis

the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis

nappe

a nappe is one half of a double cone

polar axis
the horizontal axis in the polar coordinate system corresponding to [latex]r\ge 0[/latex]

polar coordinate system

a system for locating points in the plane. The coordinates are [latex]r[/latex], the radial coordinate, and [latex]\theta[/latex], the angular coordinate

polar equation

an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system

pole

the central point of the polar coordinate system, equivalent to the origin of a Cartesian system

radial coordinate

[latex]r[/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole

rose

graph of the polar equation [latex]r=a\cos{n}\theta[/latex] or [latex]r=a\sin{n}\theta[/latex] for a positive constant [latex]a[/latex] and an integer [latex]n \ge 2[/latex]

space-filling curve

a curve that completely occupies a two-dimensional subset of the real plane

standard form

an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes

vertex

a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch