Rotation of 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 the figure below.

Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in the figure below. 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 $A,B,$ and $C$ 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 $xy$ term that we have not seen in any of the standard form equations. As we will discuss later, the $xy$ term rotates the conic whenever $B$ is not equal to zero.

Example. Identify the graph of each of the following nondegenerate conic sections.

1. $4x^2-9y^2+36x+36y-125=0$
2. $9y^2+16x+36y-10=0$
3. $3x^2+3y^2-2x-6y-4=0$
4. $-25x^2-4y^2+100x+16y+20=0$

Solution. Rewriting each general form, we identify $A$ and $C$.

1. $4x^2+0xy+(-9)y^2+36x+36y+(-125)=0$, so $A=4$ and $C=-9$. Because $A$ and $C$ have opposite signs, the graph of this equation is a hyperbola.
2. $0x^2+0xy+9y^2+16x+36y+(-10)=0$, so $A=0$ and $C=9$. We can determine that the equation is a parabola, since $A$ is zero.
3. $3x^2+0xy+3y^2+(-2)x+(-6)y+(-4)=0$, so $A=3$ and $C=3$. Because $A=C$, the graph of this equation is a circle.
4. $(-25)x^2+0xy+(-4)y^2+100x+16y+20=0$, so $A=-25$ and $C=-4$. Because $AC>0$ and $A\ne C$, the graph of this equation is an 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 an $xy$ term, which aligns the graphs with the x- and y-axes. When we add an $xy$ term, we are rotating the conic about the origin. If the x- and y-axes are rotated through an angle, say $\theta,$ then every point on the plane may be thought of as having two representations: $(x,y)$ on the Cartesian plane with the original x-axis and y-axis, and $(x',y')$ on the new plane defined by the new, rotated axes, called the x′-axis and y′-axis. See the figure below.

We will find the relationships between $x$ and $y$ on the Cartesian plane with $x'$ and $y'$ on the new rotated plane. See the figure below.

The original coordinate x- and y-axes have unit vectors $i$ and $j.$ The rotated coordinate axes have unit vectors $i'$ and $j'.$ The angle $\theta$ is known as the angle of rotation. See the figure below. We may write the new unit vectors in terms of the original ones.

Consider a vector $u$ in the new coordinate plane. It may be represented in terms of its coordinate axes.

Because $u=x'i'+y'j',$ we have representations of $x$ and $y$ in terms of the new coordinate system.

Example. Find a new representation of the equation $2x^2-xy+2y^2-30=0$ after rotating through an angle of $\theta=45^\circ.$

Solution. Find $x$ and $y,$ where $x=x'\cos\theta-y'\sin\theta$ and $y=x'\sin\theta+y'\cos\theta.$ Because $\theta=45^\circ,$

and

Substitute $x=\tfrac{x'-y'}{\sqrt2}$ and $y=\tfrac{x'+y'}{\sqrt2}$ into $2x^2-xy+2y^2-30=0.$

Simplify.

Write the equations with $x'$ and $y'$ in the standard form.

This equation is an ellipse. The figure below 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 form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the $x'$ and $y'$ coordinate system without the $x'y'$ term, by rotating the axes by a measure of $\theta$ that satisfies

We have learned already that any conic may be represented by the second degree equation

where $A,B,$ and $C$ are not all zero. However, if $B\ne0,$ then we have an $xy$ term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle $\theta$ where $\cot(2\theta)=\tfrac{A-C}{B}.$

• If $\cot(2\theta)>0,$ then $2\theta$ is in the first quadrant, and $\theta$ is between $(0^\circ,45^\circ).$
• If $\cot(2\theta)<0,$ then $2\theta$ is in the second quadrant, and $\theta$ is between $(45^\circ,90^\circ).$
• If $A=C,$ then $\theta=45^\circ.$

Example. Rewrite the equation $8x^2-12xy+17y^2=20$ in the $x'y'$ system without an $x'y'$ term.

Solution. First, we find $\cot(2\theta).$ See the figure below.

So the hypotenuse is

Next, we find $\sin\theta$ and $\cos\theta.$

Substitute the values of $\sin\theta$ and $\cos\theta$ into $x=x'\cos\theta-y'\sin\theta$ and $y=x'\sin\theta+y'\cos\theta.$

and

Substitute the expressions for $x$ and $y$ into the given equation, and then simplify.

Write the equations with $x'$ and $y'$ in the standard form with respect to the new coordinate system.

The figure below shows the graph of the ellipse.

Example. Graph the following equation relative to the $x'y'$ system: $x^2+12xy-4y^2=30.$

Solution. First, we find $\cot(2\theta).$

Because $\cot(2\theta)=\tfrac{5}{12},$ we can draw a reference triangle as in the figure below.

Thus, the hypotenuse is

Next, we find $\sin\theta$ and $\cos\theta.$ We will use half-angle identities.

Now we find $x$ and $y.$

and

Now we substitute $x=\tfrac{3x'-2y'}{\sqrt{13}}$ and $y=\tfrac{2x'+3y'}{\sqrt{13}}$ into $x^2+12xy-4y^2=30.$

The figure below shows the graph of the hyperbola $\tfrac{x'^2}{6}-\tfrac{4y'^2}{15}=1.$

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 $B^2-4AC=B'^2-4A'C'.$ The expression does not vary after rotation, so we call the expression invariant. The discriminant, $B^2-4AC,$ is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.

Example. Identify the conic for each of the following without rotating axes.

1. $5x^2+2\sqrt3xy+2y^2-5=0$
2. $5x^2+2\sqrt3xy+12y^2-5=0$

Solution. Let’s begin by determining $A,B,$ and $C.$

1. $A=5,\ B=2\sqrt3,\ C=2.$ Now, we find the discriminant.

   $$\begin{array}{lrcl} & B^2-4AC &=& (2\sqrt3)^2-4(5)(2) \\[4pt] & &=& 4(3)-40 \\[4pt] & &=& 12-40 \\[4pt] & &=& -28<0 \end{array}$$

   Therefore, $5x^2+2\sqrt3xy+2y^2-5=0$ represents an ellipse.

2. $A=5,\ B=2\sqrt3,\ C=12.$ Again, we find the discriminant.

   $$\begin{array}{lrcl} & B^2-4AC &=& (2\sqrt3)^2-4(5)(12) \\[4pt] & &=& 4(3)-240 \\[4pt] & &=& 12-240 \\[4pt] & &=& -228<0 \end{array}$$

   Therefore, $5x^2+2\sqrt3xy+12y^2-5=0$ also represents an ellipse.

Key equations

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 form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ where $A,B$ and $C$ are not all zero. The values of $A,B,$ and $C$ determine the type of conic.
• Equations of conic sections with an $xy$ term have been rotated about the origin.
• The general form can be transformed into an equation in the $x'$ and $y'$ coordinate system without the $x'y'$ term.
• 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.

Key terms

angle of rotation — an acute angle formed by a set of axes rotated from the Cartesian plane where, if $\cot(2\theta)>0,$ then $\theta$ is between $(0^\circ,45^\circ)$; if $\cot(2\theta)<0,$ then $\theta$ is between $(45^\circ,90^\circ)$; and if $\cot(2\theta)=0,$ then $\theta=45^\circ$. 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.

Practice

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

This section is adapted from Precalculus 2e, Section 10.4: Rotation of Axes by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted a coreq-skills block the pinned CNXML prepends before the section proper (its own “Objective 1”/“Objective 2” corequisite review of the rotation-of-axes substitution and of identifying conics from general form, each with its own worked example and “Practice Makes Perfect” exercise set, tagged IA 11.4.3) — confirmed against the rendered PDF, page 1056 (true PDF index 1066), where exercise 70 of the previous section’s Real-World Applications runs directly into the printed “10.4 Rotation of Axes” heading and objectives list with no corequisite-skills material between them; the same prepended-block pattern is already logged in this book’s errata for §§4.3–4.8, §§9.1–9.7, and §9.8, and this section joins that list. The section’s own four learning objectives, from the module’s abstract, are unaffected. Kept the “Media” callout’s introductory sentence but omitted its external video link, matching house precedent elsewhere in this book. Recreated all of the module’s instructional figures as accessible spec-first SVGs. The two double-cone overview figures (the nondegenerate conic sections, and the degenerate ones) are drawn as exact schematics split into two figures each to keep their labels readable: a right double cone and its cutting plane, with the base circles, the cone silhouettes, and each conic trace (ellipse, circle, hyperbola, parabola; two intersecting lines, one line, a point) computed from the cone equation under a fixed oblique projection, hidden portions dashed, the slice name above and the resulting plane curve drawn beneath each cone. The rotated-axes overview figure and the two unit-vector figures (introducing $\theta$, then $i,j,i',j'$) use an illustrative, unlabeled angle of $40^\circ$ for the unit-vector pair and $45^\circ$ for the overview, since the source art itself does not tie those diagrams to any numbered example. Every other figure — the four rotated-conic graphs (the introductory ellipse $x^2+y^2-xy-15=0$; Example 2’s ellipse $\tfrac{x'^2}{20}+\tfrac{y'^2}{12}=1$ at $\theta=45^\circ$; Example 3’s ellipse $\tfrac{x'^2}{4}+\tfrac{y'^2}{1}=1$; Example 4’s hyperbola $\tfrac{x'^2}{6}-\tfrac{4y'^2}{15}=1$) and the two right-triangle reference diagrams (legs $3,4,5$ and $5,12,13$) — is plotted from the exact algebra worked in the adjacent example, independently re-derived (including by running the rotation-of-coefficients substitution in Node) rather than traced from the source art; a rotated conic has no closed-form drawing primitive, so each is sampled from its own parametric equations into a polylines path per this book’s established recipe for rotated and polar curves. Primed variables are graded as of this authoring run: every $x'y'$-system answer is keyed with a plain apostrophe (x'^2, never x^{\prime}), and every “write the equation in the $x'y'$ system, in standard form” fillin declares answerForm="conic-standard-form" — confirmed by replaying each printed general-form equation through the grader under that form and getting form, never correct. Two Practice fillins ask for “a new representation” rather than “standard form” and carry no form token, because their resulting equations (kept as printed/derived, 7x'^2+9y'^2-4=0 and 3x'^2+2x'y'-5y'^2+1=0) cannot be written with a coefficient-1 squared term over an integer denominator; both were confirmed not retype-passable from their printed general-form subject, since the primed and unprimed symbols grade as distinct variables. The rotation-angle fillin is keyed answerForm="degrees" matching the source’s own degree unit. “Identify the conic” and the two discriminant-sign conceptual questions are multiplechoice over the conic names or their own printed explanations, since a category is never a gradable expression; the discriminant VALUE itself (Practice, objective 4) is a plain number fill-in. The four end-of-section “identify the conic” items with an $xy$ term whose Answer Key states “$B^2-4AC=\ldots$” are transcribed as the discriminant’s numeric value or the conic name depending on which the item’s own solution emphasizes. Twelve selected end-of-section exercises, adapted into thirteen Practice components (three general-form identification items; three Verbal items recast as conceptual multiple choice; two “find a new representation” items; one “eliminate the $xy$ term” item split into its angle and its resulting equation; the graph-only rewrite item discussed below; and two discriminant items, one identifying the conic and one asking for the discriminant’s value) plus the section’s own three Try Its (one two-part, one single, one two-part) were adapted into interactive components, one Practice group per objective, every one independently re-derived by computation (including by running the rotation substitution and discriminant arithmetic in Node) rather than read off the source key. One end-of-section item (rewrite $x^2-10xy+y^2-24=0$ without the $x'y'$ term) prints only a solution graph in the Answer Key, whose description is self-inconsistent — it states a rotation of $\theta=45^\circ$ yet places the vertices at $(\pm2,0)$ on the $x'$-axis, which is where the other $45^\circ$ diagonal puts them — so the question pins $\theta=45^\circ$ and keys the result of the section’s own rotation formulas, $\tfrac{y'^2}{4}-\tfrac{x'^2}{6}=1$ (the same hyperbola, vertices on the $y'$-axis), derived independently.