Analytic Geometry
64 Rotation of Axes
Learning Objectives
In this section, you will:
- Identify nondegenerate conic sections given their general form equations.
- Use rotation of axes formulas.
- Write equations of rotated conics in standard form.
- Identify conics without rotating axes.
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 (Figure).
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in (Figure). 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 anterm that we have not seen in any of the standard form equations. As we will discuss later, theterm rotates the conic wheneveris not equal to zero.
Conic Sections | Example |
---|---|
ellipse | |
circle | |
hyperbola | |
parabola | |
one line | |
intersecting lines | |
parallel lines | |
a point | |
no graph |
A conic section has the general form
where andare not all zero.
(Figure) summarizes the different conic sections where andandare nonzero real numbers. This indicates that the conic has not been rotated.
ellipse | |
circle | |
hyperbola | whereandare positive |
parabola |
Given the equation of a conic, identify the type of conic.
- Rewrite the equation in the general form,
- Identify the values ofandfrom the general form.
- Ifand are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.
- Ifandare equal and nonzero and have the same sign, then the graph may be a circle.
- Ifandare nonzero and have opposite signs, then the graph may be a hyperbola.
- If eitheror is zero, then the graph may be a parabola.
If B = 0, the conic section will have a vertical and/or horizontal axes. If B does not equal 0, as shown below, the conic section is rotated. Notice the phrase “may be” in the definitions. That is because the equation may not represent a conic section at all, depending on the values of A, B, C, D, E, and F. For example, the degenerate case of a circle or an ellipse is a point:
when A and B have the same sign.
The degenerate case of a hyperbola is two intersecting straight lines: when A and B have opposite signs.
On the other hand, the equation, when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.
Identify the graph of each of the following nondegenerate conic sections.
- Rewriting the general form, we have
and so we observe thatandhave opposite signs. The graph of this equation is a hyperbola.
- Rewriting the general form, we have
andWe can determine that the equation is a parabola, sinceis zero.
- Rewriting the general form, we have
andBecause the graph of this equation is a circle.
- Rewriting the general form, we have
andBecauseand the graph of this equation is an ellipse.
Identify the graph of each of the following nondegenerate conic sections.
- hyperbola
- ellipse
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 anterm, which aligns the graphs with the x– and y-axes. When we add anterm, we are rotating the conic about the origin. If the x– and y-axes are rotated through an angle, saythen 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, andon the new plane defined by the new, rotated axes, called the x’-axis and y’-axis. See (Figure).
We will find the relationships betweenandon the Cartesian plane withandon the new rotated plane. See (Figure).
The original coordinate x– and y-axes have unit vectorsandThe rotated coordinate axes have unit vectorsandThe angleis known as the angle of rotation. See (Figure). We may write the new unit vectors in terms of the original ones.
Consider a vectorin the new coordinate plane. It may be represented in terms of its coordinate axes.
Because we have representations ofandin terms of the new coordinate system.
If a pointon the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an anglefrom the positive x-axis, then the coordinates of the point with respect to the new axes areWe can use the following equations of rotation to define the relationship betweenand
and
Given the equation of a conic, find a new representation after rotating through an angle.
- Findandwhereand
- Substitute the expression forandinto in the given equation, then simplify.
- Write the equations withandin standard form.
Find a new representation of the equationafter rotating through an angle of
Findandwhereand
Because
and
Substituteandinto
Simplify.
Write the equations withandin the standard form.
This equation is an ellipse. (Figure) shows the graph.
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 forminto standard form by rotating the axes. To do so, we will rewrite the general form as an equation in theandcoordinate system without theterm, by rotating the axes by a measure ofthat satisfies
We have learned already that any conic may be represented by the second degree equation
whereandare not all zero. However, if then we have anterm that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute anglewhere
- If then is in the first quadrant, and is between
- If then is in the second quadrant, and is between
- If then
Given an equation for a conic in thesystem, rewrite the equation without theterm in terms ofandwhere theandaxes are rotations of the standard axes bydegrees.
- Find
- Findand
- Substituteandintoand
- Substitute the expression forandinto in the given equation, and then simplify.
- Write the equations withandin the standard form with respect to the rotated axes.
Rewrite the equationin thesystem without anterm.
First, we findSee (Figure).
So the hypotenuse is
Next, we find and
Substitute the values ofandintoand
and
Substitute the expressions forandinto in the given equation, and then simplify.
Write the equations withandin the standard form with respect to the new coordinate system.
(Figure) shows the graph of the ellipse.
Rewrite thein thesystem without theterm.
Graph the following equation relative to thesystem:
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 thatThe 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.
If the equationis transformed by rotating axes into the equation then
The equationis an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.
If the discriminant,is
- the conic section is an ellipse
- the conic section is a parabola
- the conic section is a hyperbola
Identify the conic for each of the following without rotating axes.
- Let’s begin by determining and
Now, we find the discriminant.
Therefore,represents an ellipse.
- Again, let’s begin by determining and
Now, we find the discriminant.
Therefore,represents an ellipse.
Identify the conic for each of the following without rotating axes.
- hyperbola
- ellipse
Access this online resource for additional instruction and practice with conic sections and rotation of axes.
Key Equations
General Form equation of a conic section | |
Rotation of a conic section | |
Angle of rotation |
Key Concepts
- 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.
- A nondegenerate conic section has the general formwhereandare not all zero. The values of anddetermine the type of conic. See (Figure).
- Equations of conic sections with anterm have been rotated about the origin. See (Figure).
- The general form can be transformed into an equation in theandcoordinate system without theterm. See (Figure) and (Figure).
- 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 (Figure).
Section Exercises
Verbal
What effect does theterm have on the graph of a conic section?
Theterm causes a rotation of the graph to occur.
If the equation of a conic section is written in the formand what can we conclude?
If the equation of a conic section is written in the formand what can we conclude?
The conic section is a hyperbola.
Given the equation what can we conclude if
For the equation the value ofthat satisfiesgives us what information?
It gives the angle of rotation of the axes in order to eliminate theterm.
Algebraic
For the following exercises, determine which conic section is represented based on the given equation.
parabola
hyperbola
ellipse
parabola
parabola
ellipse
For the following exercises, find a new representation of the given equation after rotating through the given angle.
For the following exercises, determine the anglethat will eliminate theterm and write the corresponding equation without theterm.
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 thesystem in which the equation has noterm.
For the following exercises, determine the angle of rotation in order to eliminate theterm. Then graph the new set of axes.
For the following exercises, determine the value ofbased on the given equation.
Given findfor the graph to be a parabola.
Given findfor the graph to be an ellipse.
Given findfor the graph to be a hyperbola.
Given findfor the graph to be a parabola.
Given findfor the graph to be an ellipse.
Glossary
- angle of rotation
- an acute angle formed by a set of axes rotated from the Cartesian plane where, ifthenis betweenifthenis betweenand ifthen
- degenerate conic sections
- any of the possible shapes formed when a plane intersects a double cone through the apex. Types of degenerate conic sections include a point, a line, and intersecting lines.
- nondegenerate conic section
- a shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas