12.4 Rotation of Axes

Conic Sections

General form of conic sections

Every conic section can be written in a single general equation:

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

The coefficients $A$, $B$, and $C$ can't all be zero at the same time (otherwise you'd just have a linear equation or a constant).

The discriminant $B^2 - 4AC$ tells you which type of conic you're dealing with:

• Ellipse when $B^2 - 4AC < 0$ (a closed oval shape)
• Parabola when $B^2 - 4AC = 0$ (a U-shaped curve that opens in one direction)
• Hyperbola when $B^2 - 4AC > 0$ (two separate curved branches)

A circle is a special case of an ellipse where $B = 0$ and $A = C$. The discriminant is invariant under rotation, so it gives the same classification no matter how the axes are oriented.

Rotation of axes for conics

When the general equation has a $Bxy$ term ($B \neq 0$), the conic is tilted relative to the $x$- and $y$-axes. Rotation of axes is the technique for eliminating that $xy$ term so you can work with a simpler equation in a new coordinate system $(x', y')$.

Finding the rotation angle:

The angle $\theta$ that eliminates the $xy$ term satisfies:

$\cot 2\theta = \frac{A - C}{B}$

Choose $\theta$ so that $0° < \theta < 90°$. If $A = C$, then $\cot 2\theta = 0$, which means $\theta = 45°$.

Substitution formulas:

Once you have $\theta$, replace $x$ and $y$ using:

• $x = x'\cos\theta - y'\sin\theta$
• $y = x'\sin\theta + y'\cos\theta$

Carrying out the rotation step by step:

1. Start with the general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

2. Compute $\theta$ from $\cot 2\theta = \frac{A - C}{B}$. Use a half-angle identity or right-triangle approach to find $\cos\theta$ and $\sin\theta$.

3. Substitute the rotation formulas for $x$ and $y$ into the equation.

4. Expand every term and collect like terms in $x'^2$, $x'y'$, $y'^2$, $x'$, $y'$, and the constant.

5. The $x'y'$ coefficient should simplify to zero. You're left with an equation in $x'$ and $y'$ with no cross term.

Standard form of rotated conics

After rotation, your equation looks like:

$A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0$

To put this into standard form (so you can read off the center, vertices, etc.), complete the square:

1. Move the constant $F'$ to the right side of the equation.
2. Group the $x'$ terms together and the $y'$ terms together.
3. Factor out the leading coefficient from each group. For example, factor $A'$ out of the $x'$ terms.
4. Complete the square inside each group: take half the linear coefficient, square it, and add it to both sides (accounting for the factored-out coefficient).
5. Rewrite each group as a squared binomial and simplify the right side.

The result is a standard-form equation you can use to identify the center, axes, vertices, and other features of the conic in the rotated coordinate system.

Analysis of non-rotated conics

Once the $xy$ term is gone (either because it was never there, or because you rotated it away), you identify the conic type using the discriminant $B^2 - 4AC$ and then extract its geometric features.

Ellipses and hyperbolas:

• Find the center $(h, k)$ by completing the square for both variables.
• The denominators in the standard form give you the lengths of the axes. For an ellipse, the larger denominator corresponds to the semi-major axis and the smaller to the semi-minor axis.
• Eccentricity measures how much the conic deviates from a circle: $e = 0$ for a circle, $0 < e < 1$ for an ellipse, $e = 1$ for a parabola, and $e > 1$ for a hyperbola.

Parabolas:

• Find the vertex by completing the square for the variable that's squared.
• The sign of the squared term's coefficient tells you the direction of opening (positive coefficient on $y^2$ opens right; negative opens left; positive on $x^2$ opens up; negative opens down).
• Use the standard form to locate the focus and directrix, which sit at equal distances from the vertex along the axis of symmetry.

Mathematical Foundations

• Coordinate plane: The 2D system where conic sections are graphed. Rotation of axes defines a new coordinate grid tilted by angle $\theta$ relative to the original.
• Quadratic equations: The general second-degree equation is the algebraic backbone of all conic sections.
• Trigonometry: Half-angle and double-angle identities are essential for computing $\cos\theta$ and $\sin\theta$ from the rotation angle formula.