Recipe

Rotate away the cross term (Step 2 continued, when B ≠ 0)

Use when: Step 2 of the general-form recipe found B\neq0, so the conic is rotated and needs its axes turned before Steps 3 and 4 can read the equation

Goal: a new equation in x' and y', with no x'y' term, ready to hand back to Step 3

Taught in Rotation of axes · extends Analyze a conic from general form

1. 1

   Solve \cot2\theta=\frac{A-C}{B} for \theta between 0° and 90°.

   Example

   A=5, B=-6, C=5: \cot2\theta=\frac{5-5}{-6}=0, so 2\theta=90° and \theta=45°.

   Why: this is the one angle that makes the rotated equation's cross-term coefficient vanish, the whole point of rotating at all.

2. 2

   Replace x and y with x=x'\cos\theta-y'\sin\theta and y=x'\sin\theta+y'\cos\theta, then expand and combine like terms.

   Example

   At \theta=45°, 5x^2-6xy+5y^2=8 becomes 2x'^2+8y'^2=8, no x'y' term left -- an ellipse, \frac{x'^2}{4}+y'^2=1 (the same equation the shortcut below classifies without fully rotating).

   Why: the double-angle identities guarantee the x'y' coefficient is exactly zero at this angle; that's what \cot2\theta=\frac{A-C}{B} was solved for.

3. 3

   The new equation is in x', y' with B=0. Pick up the general-form recipe at Step 3, reading the family from the new A' and C'.

   Why: rotating never changes the conic's shape or size, only which letters name its axes. Every later step works exactly as before, just primed.