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NY Regents Geometry Conic Sections in Polar Coordinates — practice questions

Explore 4 free Conic Sections in Polar Coordinates questions in NY Regents Geometry Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Conic Sections in Polar Coordinates questions and exercises with answers

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A conic written in polar form has eccentricity equal to 1. Which curve is it?Read the answer and worked solution →Rewriting r equals 6 over the quantity 2 plus 3 cos theta in standard form gives which eccentricity and curve?Read the answer and worked solution →Eccentricity is the ratio of which two distances measured from a point on the curve?Read the answer and worked solution →Cluster the conics by eccentricityRead the answer and worked solution →

Worked practice questions

A conic written in polar form has eccentricity equal to 1. Which curve is it?

  1. An ellipse
  2. A hyperbola
  3. A circle
  4. A parabola
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Correct answer: D. A parabola

1. Eccentricity compares the distance to the focus with the distance to the directrix. 2. A value below 1 gives an ellipse and a value above 1 gives a hyperbola. 3. The single value e equal to 1 marks the parabola, the boundary between the two. _Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.5 Conic Sections in Polar Coordinates_

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Rewriting r equals 6 over the quantity 2 plus 3 cos theta in standard form gives which eccentricity and curve?

  1. e is 1.5, so a hyperbola
  2. e is 3.0, so a hyperbola
  3. e is 0.5, so an ellipse
  4. e is 1.0, so a parabola
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Correct answer: A. e is 1.5, so a hyperbola

1. Standard form needs the denominator to begin with 1, so divide top and bottom by 2. 2. The numerator 6 divided by 2 becomes 3. 3. The denominator 2 plus 3 cos theta divided by 2 becomes 1 plus 1.5 cos theta. 4. The eccentricity is the coefficient sitting in front of the cosine, so e is 1.5. 5. Since 1.5 is greater than 1, the curve is a hyperbola. 6. Reading e as 3.0 takes the original coefficient without dividing the denominator through. 7. Reading e as 0.5 inverts the ratio and would wrongly promise a closed curve. 8. Reading e as 1.0 would require the cosine coefficient to match the leading 1 exactly. _Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.5 Conic Sections in Polar Coordinates_

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Eccentricity is the ratio of which two distances measured from a point on the curve?

  1. To the vertex over to the focus
  2. To the focus over to the directrix
  3. To the center over to the focus
  4. To the directrix over to the pole
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Correct answer: B. To the focus over to the directrix

1. Pick any point P on the curve, with F the focus and D the directrix. 2. Measure the distance from P to F and the distance from P to D. 3. Eccentricity is the first divided by the second, and it is the same for every P. 4. Using the vertex or the center in place of the directrix gives a ratio that changes from point to point. _Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.5 Conic Sections in Polar Coordinates_

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More Mathematics topics

Law of CosinesThe EllipseConic Sections in General FormThe ParabolaLaw of SinesThe HyperbolaVectorsGraphs of Polar Equations