# Analytic Geometry

## Rotation of Axes


### Learning Objectives

1. Using rotation of axes formulas.
2. Identify conic sections by their equations. (IA 11.4.3)

### Objective 1: Using rotation of axes formulas.

If a point
 

on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle
 

from the positive
x
-axis, then the coordinates of the point with respect to the new axes are
 


The following rotations of axes formulas define the relationship between (x,y) and (x’,y’):

### Practice Makes Perfect

Using rotation of axes formulas:

### Objective 2: Identify conic sections by their equations. (IA 11.4.3)


We can identify a conic from its equations by looking at the signs and coefficients of the variables that are squared.


### Practice Makes Perfect


Identify conic sections by their equations.

As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See .
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in . A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines. 

### Identifying Nondegenerate Conics in General Form

In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below. 
where 
    and 
    are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.
You may notice that the general form equation has an 
    term that we have not seen in any of the standard form equations. As we will discuss later, the 
    term rotates the conic whenever 
  is not equal to zero. 

### Finding a New Representation of the Given Equation after Rotating through a Given Angle

Until now, we have looked at equations of conic sections without an 
     term, which aligns the graphs with the x- and y-axes. When we add an 
     term, we are rotating the conic about the origin. If the x-  and y-axes are rotated through an angle, say 
     then every point on the plane may be thought of as having two representations: 
     on the Cartesian plane with the original x-axis and y-axis, and 
  on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis. See .
We will find the relationships between 
     and 
     on the Cartesian plane with 
     and 
     on the new rotated plane. See .
The original coordinate x- and y-axes have unit vectors 
     and 
     The rotated coordinate axes have unit vectors 
     and 
     The angle 
     is known as the angle of rotation. See . We may write the new unit vectors in terms of the original ones.
Consider a vector in the new coordinate plane. It may be represented in terms of its coordinate axes.
Because 
     we have representations of 
     and 
     in terms of the new coordinate system.

### Writing Equations of Rotated Conics in Standard Form

Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form 
    into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the 
    and 
    coordinate system without the 
    term, by rotating the axes by a measure of 
    that satisfies
We have learned already that any conic may be represented by the second degree equation
where 
    and 
    are not all zero. However, if 
    then we have an 
    term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle 
    where 
   

  

###  Identifying Conics without Rotating Axes

Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is
If we apply the rotation formulas to this equation we get the form
It may be shown that 
    The expression does not vary after rotation, so we call the expression invariant. The discriminant, 
    is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.

### Key Equations


### Key Concepts

1. Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
2. A nondegenerate conic section has the general form 
  	   where 
  	   and 
  	   are not all zero. The values of 
  	   and 
  	   determine the type of conic. See .
3. Equations of conic sections with an 
  	   term have been rotated about the origin. See .
4. The general form can be transformed into an equation in the 
  	   and 
  	   coordinate system without the 
  	   term. See  and .
5. An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, determine which conic section is represented based on the given equation.
For the following exercises, find a new representation of the given equation after rotating through the given angle.
For the following exercises, determine the angle 
     that will eliminate the 
     term and write the corresponding equation without the 
     term.

### Graphical

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.
For the following exercises, graph the equation relative to the 
     system in which the equation has no 
     term.
For the following exercises, determine the angle of rotation in order to eliminate the 
     term. Then graph the new set of axes.
For the following exercises, determine the value of 
     based on the given equation.