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11.5 Rotation of Axes

Focus on new-coordinate formula and conic discriminant.

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01Rotation of axes formulasWorked steps · five questions 02Eliminating the xy-term and the discriminantWorked steps · five questions

Lesson brief and rules

11.5 Rotation of Axes

For axes rotated counterclockwise by $\phi , X=x \cos \phi +y \sin \phi$ and $Y=-x \sin \phi +y \cos \phi$ . Use $\cot (2\phi )=\frac{A-C}{B}$ to eliminate the $xy-$ term.

Worked example

1

01 / The coordinate axes rotate counterclockwise by

\phi

while the point stays fixed. Which formula gives

X

02 / Use

x = X \cos \phi - Y \sin \phi

and

y = X \sin \phi + Y \cos \phi

. Inverting gives

X = x \cos \phi + y \sin \phi

03 / Answer:

X = x \cos \phi + y \sin \phi

Worked example 2 · Conic discriminant

For $4x^{2} + 5xy - 6y^{2} = 1$ , find $B^{2} - 4AC$ and classify the conic.

$A = 4, B = 5, C = -6$ . $B^{2} - 4AC = 25 + 96 = 121 > 0$ , so the nondegenerate conic is a hyperbola. Answer: $121$ ; hyperbola

15 questions covering every idea, with a worked answer for each.Start practice ↗

Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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11.5 Rotation of Axes

Ideas in this lesson

11.5 Rotation of Axes