Lesson 12.4: Rotation of Axes

New Concept

The $xy$ term in a conic's equation means its graph is rotated. You'll learn to find the angle of rotation, $\theta$, to transform the equation, remove the $xy$ term, and identify the conic's true shape.

What’s next

Next, you'll dive into interactive examples using rotation formulas. You'll then apply what you've learned on practice cards and challenge problems.

Property

A conic section has the general form $A x^2 + B x y + C y^2 + D x + E y + F = 0$ where $A$, $B$, and $C$ are not all zero. To identify the type of conic when it has not been rotated ($B=0$):

1. Rewrite the equation in the general form.
2. Identify the values of $A$ and $C$.

a. If $A$ and $C$ are nonzero, have the same sign, and are not equal to each other ($AC > 0, A \neq C$), the graph may be an ellipse.
b. If $A$ and $C$ are equal and nonzero ($A = C$), the graph may be a circle.
c. If $A$ and $C$ are nonzero and have opposite signs ($AC < 0$), the graph may be a hyperbola.
d. If either $A$ or $C$ is zero, the graph may be a parabola.

Examples

• For the equation $5x^2 - 3y^2 + 10x - 12y + 7 = 0$, we see that $A=5$ and $C=-3$. Since $A$ and $C$ have opposite signs, the graph is a hyperbola.
• For the equation $7x^2 + 7y^2 - 14x + 28y - 50 = 0$, we see that $A=7$ and $C=7$. Since $A=C$, the graph is a circle.
• For the equation $4x^2 + 20x - 3y + 1 = 0$, we see that $A=4$ and $C=0$. Since $C$ is zero, the graph is a parabola.

Explanation

Think of coefficients $A$ and $C$ as clues to the conic's identity. By comparing their signs and values in the general form, you can quickly determine the shape of the conic section, as long as it has not been rotated.

Property

If a point $(x, y)$ on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle $\theta$ from the positive $x$-axis, then the coordinates of the point with respect to the new axes are $(x', y')$. We can use the following equations of rotation to define the relationship between $(x, y)$ and $(x', y')$:

and

Examples

• To find the new equation for $x=5$ after a rotation of $\theta=30^\circ$, we substitute $x = x' \cos(30^\circ) - y' \sin(30^\circ)$. This gives $x'(\frac{\sqrt{3}}{2}) - y'(\frac{1}{2}) = 5$, or $\sqrt{3}x' - y' = 10$.
• Find the new representation of $y=x$ after rotating by $\theta = 45^\circ$. Substitute $x = \frac{x' - y'}{\sqrt{2}}$ and $y = \frac{x' + y'}{\sqrt{2}}$. The equation becomes $\frac{x' + y'}{\sqrt{2}} = \frac{x' - y'}{\sqrt{2}}$, which simplifies to $y'=0$.
• Find the new equation for the circle $x^2 + y^2 = 16$ after a $60^\circ$ rotation. Substituting the rotation formulas for $x$ and $y$ and simplifying gives $(x')^2 + (y')^2 = 16$. A circle centered at the origin looks the same after any rotation.

Explanation

These formulas act as a translator between the original coordinate system and a new, rotated one. By substituting these expressions for $x$ and $y$ into an equation, we can see what the equation looks like in the rotated view.

Property

To transform the equation of a conic $A x^2 + B x y + C y^2 + D x + E y + F = 0$ into an equation in the $x'$ and $y'$ coordinate system without the $x'y'$ term, we rotate the axes by a measure of $\theta$ that satisfies

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 $\theta = 45^\circ$.

Examples

• For the equation $5x^2 - 8xy + 11y^2 = 15$, we have $A=5, B=-8, C=11$. The angle of rotation $\theta$ is found using $\cot(2\theta) = \frac{5-11}{-8} = \frac{-6}{-8} = \frac{3}{4}$.
• For the equation $3x^2 + 2xy + 3y^2 = 10$, we have $A=3, B=2, C=3$. Since $A=C$, the angle of rotation is $\theta = 45^\circ$.
• For the equation $x^2 + 4xy - 2y^2 = 9$, we have $A=1, B=4, C=-2$. The angle of rotation $\theta$ is found using $\cot(2\theta) = \frac{1 - (-2)}{4} = \frac{3}{4}$.

Explanation

The $xy$ term indicates a rotated conic. This formula is the key to finding the precise angle of rotation needed to align the conic with the new $x'$ and $y'$ axes, which makes the $x'y'$ term disappear and simplifies the equation.

Property

If the equation $A x^2 + B x y + C y^2 + D x + E y + F = 0$ is transformed by rotating axes into the equation $A' x'^2 + B' x' y' + C' y'^2 + D' x' + E' y' + F' = 0$, then $B^2 - 4AC = B'^2 - 4A'C'$. The equation is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these. If the discriminant, $B^2 - 4AC$, is

• $< 0$, the conic section is an ellipse
• $= 0$, the conic section is a parabola
• $> 0$, the conic section is a hyperbola

Examples

• For $x^2 + 3xy + 5y^2 = 10$, the discriminant is $B^2 - 4AC = (3)^2 - 4(1)(5) = 9 - 20 = -11$. Since $-11 < 0$, the conic is an ellipse.
• For $4x^2 - 4xy + y^2 + 2x = 7$, the discriminant is $B^2 - 4AC = (-4)^2 - 4(4)(1) = 16 - 16 = 0$. Since the discriminant is $0$, the conic is a parabola.
• For $2x^2 + 10xy - 3y^2 = 8$, the discriminant is $B^2 - 4AC = (10)^2 - 4(2)(-3) = 100 + 24 = 124$. Since $124 > 0$, the conic is a hyperbola.

Explanation

The discriminant, $B^2 - 4AC$, is a special value that doesn't change when a conic is rotated. This makes it a powerful shortcut to identify the conic's type without performing any rotation calculations.