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Precalculus practice

Polar Conic Sections

Normalize, classify, graph, and write focus-at-pole polar conics from eccentricity and an axis-parallel directrix.

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Normalize to r = ed/(1 plus or minus e cos(theta) or e sin(theta)), classify from e, and state the directrix and denominator restriction.

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Learning objective

What this activity builds

Connect eccentricity, directrix orientation, denominator sign, and conic classification in polar form.

Worked example

Classify r=2/(1+0.5cos(theta)).

Ellipse, e=1/2.

A positive eccentricity below 1 identifies an ellipse.

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  • Read e before graphing.
  • Relate cosine to a vertical directrix and sine to a horizontal directrix.
  • Exclude angles that make the denominator zero.

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Math SheetNameScoreTeacherDatePolar Conic SectionsNormalize to r = ed/(1 plus or minus e cos(theta) or e sin(theta)), classify from e, and state the directrix and denominator restriction.1. Graph r = 20/(1 + 2 sine(theta)) on the supplied window -147 <= x <= 147, -147 <= y <= 147 from its focus-at-pole, directrix, and eccentricity features.Assumptions: The focus is at the pole, the directrix is parallel to a coordinate axis, and denominator-zero angles are excluded rather than assigned r=0.-147-147-73.5-73.50073.573.51471472. Write the polar equation for a parabola with focus at the pole, e=1, and directrix y = 5.Assumptions: The focus is at the pole, the directrix is parallel to a coordinate axis, and denominator-zero angles are excluded rather than assigned r=0.Show exact work and preserve every stated domain restriction.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDatePolar Conic SectionsNormalize to r = ed/(1 plus or minus e cos(theta) or e sin(theta)), classify from e, and state the directrix and denominator restriction.3. Graph r = 3/(1 + 1 cosine(theta)) on the supplied window -25 <= x <= 25, -25 <= y <= 25 from its focus-at-pole, directrix, and eccentricity features.Assumptions: The focus is at the pole, the directrix is parallel to a coordinate axis, and denominator-zero angles are excluded rather than assigned r=0.-25-25-12.5-12.50012.512.525254. Write the polar equation for a hyperbola with focus at the pole, e=5/2, and directrix y = 8.Assumptions: The focus is at the pole, the directrix is parallel to a coordinate axis, and denominator-zero angles are excluded rather than assigned r=0.Show exact work and preserve every stated domain restriction.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDatePolar Conic SectionsNormalize to r = ed/(1 plus or minus e cos(theta) or e sin(theta)), classify from e, and state the directrix and denominator restriction.5. Write the polar equation for a parabola with focus at the pole, e=1, and directrix x = 10.Assumptions: The focus is at the pole, the directrix is parallel to a coordinate axis, and denominator-zero angles are excluded rather than assigned r=0.6. Write the polar equation for a ellipse with focus at the pole, e=3/4, and directrix y = -4.Assumptions: The focus is at the pole, the directrix is parallel to a coordinate axis, and denominator-zero angles are excluded rather than assigned r=0.Show exact work and preserve every stated domain restriction.math-sheet.comPage 3 of 3

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