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Warm-Up Convert the following to polar form and identify the

graph.

1. 𝑥 + 6 2 + (𝑦 − 2)2 = 40

2. 𝑥2

4−

𝑦2

9= 1

Convert the following to rectangular form and identify the graph.

1. 𝜃 =2𝜋

33. 𝑟 = −6𝑐𝑜𝑠𝜃

2. 𝑟 = −5 4. 𝜃 = −𝜋

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ConicsParabola

A parabola is the set of all points equidistant from a line and a fixed point not on the line.

The line is called the directrix, and the point is called the focus.

The point on the parabola halfway between the focus and the directrix is the vertex.

Ellipsethe set of points such that the sum of the distances to two fixed points (the foci) is constant.

Circle

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Hyperbola the set of points in a plane whose distances to two fixed points in the plane have a constant difference

Trouble in Paradise… How do I graph these equations…

Use Polar Equations!!!

Check the eccentricity, which is a measure describing how far the figure is from being a circle

Conics• To analyze a polar equation, put the equation in standard form• Then check the eccentricity

If 0 < e < 1, then the conic is an ellipseIf e = 1, then the conic is a parabolaIf e > 1, then the conic is an hyperbola

Polar EquationsDetermine the type of conic. Then graph!

Rectangular Form Identify each type of conic section. Be prepared to explain

why.

1. 𝑥 − 3 2 = 12(𝑦 − 7) 5. 𝑥 − 3 2 + 𝑦 + 4 2 = 25

2. 𝑥+2 2

9+

𝑦2

49= 1 6.

𝑦−7 2

4−

𝑥2

33= 1

3. 𝑥 − 4 2 + 𝑦 − 2 2 = 20 7. 10 𝑥 + 11 = 𝑦 + 3 2

4. 𝑥+6 2

64−

𝑦+5 2

58= 1 8.

𝑥+4 2

9+

𝑦+3 2

4= 1