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Tilted Conics

1. Use the discriminant to classify conics.2. Rotate the coordinate axes to eliminate

the xy term in equations of conics.

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Classifying Conics

Non-Degenerate tilted Degenerate

+ Hyperbola 2 intersecting lines

0 Parabola 2 || lines

- Ellipse A point

acb 42

For Ax2+Bxy+Cy2+Dx+Ey+F=0, the xy tem gives a tilt to a hyperbola, ellipse, or parabola.

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Example b2-4ac Type of Conic

x2+2xy+y2-6x+7y=17

x2+2xy+5y2-6x+7y=17

x2+12xy+5y2-6x+7y=17

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Sketching a Tilted Conic

1. Find the x and y intercepts2. Classify the conic using the discriminant3. Change into form.4. Solve for y using quadratic formula.5. Graph on calculator to classify conic and

sketch.

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𝑥2+𝑥𝑦+𝑦2−5 𝑥−36=0

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5 𝑥2−8𝑥𝑦+16 𝑦2−28 𝑥+16 𝑦+40=0

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2 𝑥2+7 𝑥𝑦+3 𝑦2+𝑥−7 𝑦−6=0

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4 𝑥2−4 𝑥𝑦+𝑦 2−2𝑥+𝑦−12=0

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𝑥2+𝑥𝑦+𝑦2=1

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Rotation of Axes

How do you eliminate the xy term from the general equation to help

classify and sketch?

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Rotation Formulas

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Find the angle of rotation needed to eliminate the xy term

• 3xy+y-2=0

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xy+2=0

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𝑥2+𝑥𝑦+𝑦2=1

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5 𝑥2−2 𝑥𝑦+5 𝑦 2−12=0

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5 𝑦 2−12=0

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Practice Day

Open Notebook Quiz