10.4Rotating Coordinate Axes
Apply rotation formulas x = x'cosθ − y'sinθ to eliminate the Bxy term. Use the discriminant B²−4AC: negative → ellipse, zero → parabola, positive → hyperbola.
Rotation of axes reveals the true shape of any conic section — even when it appears tilted. The discriminant B²−4AC is a rotation-invariant that instantly classifies any second-degree equation as an ellipse, parabola, or hyperbola.
Essential Question: How does rotating the coordinate axes eliminate the xy-term from a conic equation, and how do we use the discriminant B²−4AC to identify the type of conic without rotating?
Lesson Overview
General second-degree equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0
The xy-term (B ≠ 0) indicates the conic is rotated relative to the coordinate axes.
Rotation formulas (rotate axes by angle θ):
- x = x'cosθ − y'sinθ
- y = x'sinθ + y'cosθ
Angle of rotation to eliminate xy-term: cot(2θ) = (A−C)/B
After substitution, the new equation has no x'y' term.
Discriminant test (identifies conic type WITHOUT rotating):
- B² − 4AC < 0 → Ellipse (or circle if A = C and B = 0)
- B² − 4AC = 0 → Parabola
- B² − 4AC > 0 → Hyperbola
Invariants under rotation: A + C and B² − 4AC remain constant.
Worked Examples
Use the discriminant to identify: 2x² + 4xy + 3y² − 5x + 2y − 1 = 0.
A = 2, B = 4, C = 3
B²−4AC = 16 − 4(2)(3) = 16 − 24 = −8
B²−4AC < 0 → Ellipse
Find the angle of rotation to eliminate the xy-term in xy = 1.
Rewrite: 0·x² + 1·xy + 0·y² − 1 = 0; A = 0, B = 1, C = 0
cot(2θ) = (A−C)/B = (0−0)/1 = 0
2θ = 90° → θ = 45°
Rotate xy = 1 by 45° and write in standard form.
x = x'cos45°−y'sin45° = (x'−y')/√2
y = x'sin45°+y'cos45° = (x'+y')/√2
xy = (x'−y')(x'+y')/2 = (x'²−y'²)/2 = 1
x'²/2 − y'²/2 = 1
Identify the conic: 5x² − 6xy + 5y² − 8 = 0.
A = 5, B = −6, C = 5
B²−4AC = 36 − 4(5)(5) = 36 − 100 = −64
B²−4AC < 0 → Ellipse
cot(2θ) = (5−5)/(−6) = 0 → θ = 45°
Use the discriminant to identify: x² + 4xy + 4y² + 2x − y = 0.
A = 1, B = 4, C = 4
B²−4AC = 16 − 4(1)(4) = 16 − 16 = 0
B²−4AC = 0 → Parabola
Guided Practice
Use the discriminant to identify: 3x² − 2xy + y² + x − 4 = 0.
Hint: Compute B²−4AC with A = 3, B = −2, C = 1.
Find the angle of rotation for: x² + 2√3·xy − y² = 4.
Hint: cot(2θ) = (A−C)/B = (1−(−1))/(2√3) = 2/(2√3) = 1/√3. What angle has cot = 1/√3?
Identify: 4x² + 4xy + y² − 8x + 4y = 0.
Hint: B²−4AC = 16 − 4(4)(1) = 0. What type of conic has discriminant = 0?
For the rotation x = x'cosθ − y'sinθ, y = x'sinθ + y'cosθ with θ = 30°, express x and y in terms of x' and y'.
Hint: cos30° = √3/2, sin30° = 1/2.
Verify that B²−4AC is invariant: show that for 2x²+4xy+3y²=1, after rotating by the appropriate angle, the new equation has the same discriminant value.
Hint: After rotation, B'=0. New discriminant = 0²−4A'C' = −4A'C'. Use A+C = A'+C' and B²−4AC = B'²−4A'C'.
Key Vocabulary
General second-degree equation
Ax²+Bxy+Cy²+Dx+Ey+F=0; the xy-term indicates rotation
Rotation of axes
Transforming (x,y) to (x',y') by rotating the coordinate system by angle θ
Angle of rotation
θ chosen so that cot(2θ) = (A−C)/B, eliminating the x'y' term
Discriminant
B²−4AC; determines conic type without rotating
Invariant
A quantity unchanged by rotation; A+C and B²−4AC are invariants
Rotation formulas
x = x'cosθ−y'sinθ; y = x'sinθ+y'cosθ
Degenerate conic
A conic that reduces to a point, line, or pair of lines
Quick Check
Interactive Practice — 5 Questions
For 3x² + 5xy − 2y² + 1 = 0, the discriminant B²−4AC =
B²−4AC = 0 indicates a:
The angle of rotation to eliminate xy in x²+2xy+y²=1 is:
After rotating by 45°, xy=4 becomes:
Which quantity is NOT invariant under rotation of axes?
Independent Practice
Independent Practice
Use the discriminant to identify: x² − 3xy + 2y² + x − 1 = 0.
Find the rotation angle for: x² + xy + y² = 3.
Identify: 9x² − 24xy + 16y² + 5x − 10y = 0.
Rotate x² − xy + y² = 2 by the appropriate angle and write in standard form.
Identify: x² + 4xy − 2y² + 3x = 0.
Common Mistakes
Forgetting that B is the coefficient of xy, not x²y or xy²
In Ax²+Bxy+Cy²+Dx+Ey+F=0, B is specifically the coefficient of the xy cross-term. Identify it carefully.
Using cot(2θ) = B/(A−C) instead of (A−C)/B
The correct formula is cot(2θ) = (A−C)/B. The numerator is A−C, denominator is B.
Applying the discriminant test to equations not in general form
First write the equation as Ax²+Bxy+Cy²+Dx+Ey+F=0. Identify A, B, C correctly before computing B²−4AC.
Thinking the discriminant identifies the conic's orientation
The discriminant only identifies the TYPE (ellipse/parabola/hyperbola). It doesn't tell you the orientation or position.
Math Tips
Discriminant shortcut: B²−4AC < 0 → Ellipse, = 0 → Parabola, > 0 → Hyperbola. Memorize: 'Less is Ellipse, Zero is Parabola, Greater is Hyperbola.'
When B=0, no rotation is needed — the conic is already aligned with the axes.
For θ=45°: cos45°=sin45°=1/√2. The rotation formulas simplify to x=(x'−y')/√2, y=(x'+y')/√2.
After rotation, verify B'=0 in the new equation. If not, you made an arithmetic error.
The sum A+C is invariant: after rotation, A'+C' = A+C. Use this as a check.