Calc 2 Lecture NotesSection 9.6Page 1 of 9

Section 9.6: Conic Sections

Big idea: The conic sections are a group of graphs (the point, line, circle, parabola, ellipse, and hyperbola) that share some amazing properties:

  1. Any of the graphs can be obtained from an intersection of a double cone and a plane.
  2. The graph of any quadratic equation in two variables is one of the conic sections.
  3. The graphs can be specified as a locus of points with special distance relationships to one or two points or a line.
  4. The graphs have useful reflection properties for light and sound waves.
  5. The graphs are the trajectories of an object moving under the influence of a central force (like gravity or a Coulomb force).

Big skill: You should be able to graph a conic section given in standard form.

The conic sections obtained as the intersection of a plane with a double cone.

This picture is from:

Weisstein, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource.

Circles

A circle is the locus of all points that are a given fixed distance from a given fixed point.

Derivation of the equation of a circle:

Let (h, k) be the given point and r be the fixed distance

For any point (x, y) on the circle,
= r

Practice:

  1. Graph the equation .

Parabolas

A parabola is the locus of all points that are equidistant from a given fixed point (called the focus) and a line (called the directrix).

Derivation of the equation of a parabola:

Let (h, k) be the focus and y = d be the directrix.

For any point (x, y) on the parabola,

Theorem 6.1: Equation of a parabola and its relation to geometry

The parabola with vertex at the point (b, c), focus at (b, c + 1/(4a)), and directrix given by the line y = c – 1/(4a) is described by the equation y = a(x – b)2 + c.

Practice:

  1. Graph the equation , and state the focus and directrix.

Ellipses

Anellipse is the locus of all points the sum of whose distances from two given fixed points (called the foci) is a constant.

Derivation of the equation of an ellipse:

Let the foci be on a horizontal line and equidistant from a central point (x0, y0), and let the sum of the distances from a point on the ellipse to the foci be a constant k.

For any point (x, y) on the ellipse,

If we let and , then we get .

Theorem 6.3: Equation of an ellipse and its relation to geometry

The equation , where ab > 0 describes an ellipse with the following properties:

Center at

Foci at , or (if ba), where

Vertices at and

Practice:

  1. Graph the equation

Hyperbolas

A hyperbola is the locus of all points the difference of whose distances from two given fixed points (called the foci) is a constant.

Derivation of the equation of a hyperbola:

Let the foci be on a horizontal line and equidistant from a central point (x0, y0), and let the sum of the distances from a point on the ellipse to the foci be a constant k.

For any point (x, y) on the hyperbola,

If we let and , then we get .

Theorem 6.4: Equation of a hyperbola and its relation to geometry

The equation describes a hyperbola with the following properties:

Center at

Foci at , where

Vertices at

Asymptotes

Practice:

  1. Graph the equation .
Parabola:Fold the points on the line (directrix) onto point F (the focus) / Ellipse: Fold point F (a focus) onto the points of the circle centered at C.
Hyperbola: Fold point F (a focus) onto the points of the circle centered at C.