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Worksheet: Polar Equation of a Conic

In this worksheet, we will practice writing polar equations of conics given the eccentricity and some other characteristic, such as the equation of the directrix.

Q1:

A conic with focus at the pole has eccentricity 𝑒 = 1 2 and vertices at ( 0 , βˆ’ 6 ) and ( 0 , 2 ) . Write the equation of the conic in polar form.

  • A π‘Ÿ = 3 1 βˆ’ 0 . 5 πœƒ c o s
  • B π‘Ÿ = 3 1 βˆ’ 0 . 5 πœƒ s i n
  • C π‘Ÿ = 3 1 + 0 . 5 πœƒ c o s
  • D π‘Ÿ = 3 1 + 0 . 5 πœƒ s i n
  • E π‘Ÿ = 6 1 + 0 . 5 πœƒ s i n

Q2:

A conic with focus at the pole has eccentricity 𝑒 = 2 3 and directrix 𝑦 = βˆ’ 5 2 .

Identify the type of the conic.

  • AParabola
  • BCircle
  • CHyperbola
  • DEllipse

Write the equation of the conic in polar form.

  • A π‘Ÿ = 5 3 + 2 πœƒ c o s
  • B π‘Ÿ = 5 3 + 2 πœƒ c o s
  • C π‘Ÿ = 5 3 + 2 πœƒ s i n
  • D π‘Ÿ = 5 3 βˆ’ 2 πœƒ s i n

Q3:

A conic with its focus at the pole has eccentricity 𝑒 = 3 2 and vertices at ( βˆ’ 1 , 0 ) and ( βˆ’ 5 , 0 ) .

Identify the type of the conic.

  • AEllipse
  • BParabola
  • CCircle
  • DHyperbola

By deciding whether the directrix is in the form π‘₯ = 𝑑 , π‘₯ = βˆ’ 𝑑 , 𝑦 = 𝑑 , or 𝑦 = βˆ’ 𝑑 , where 𝑑 > 0 , select the form of the polar equation of the conic.

  • A π‘Ÿ = 𝑒 𝑑 1 + 𝑒 πœƒ s i n
  • B π‘Ÿ = 𝑒 𝑑 1 βˆ’ 𝑒 πœƒ s i n
  • C π‘Ÿ = 𝑒 𝑑 1 + 𝑒 πœƒ c o s
  • D π‘Ÿ = 𝑒 𝑑 1 βˆ’ 𝑒 πœƒ c o s

By writing one of the vertices in polar form, find the equation of the directrix.

  • A π‘₯ = βˆ’ 5 3
  • B π‘₯ = βˆ’ 2 . 5
  • C0
  • D π‘₯ = 5 3

Hence, write the equation of the conic.

  • A π‘Ÿ = 2 . 5 1 βˆ’ 1 . 5 πœƒ c o s
  • B π‘Ÿ = 2 . 5 1 + 1 . 5 πœƒ s i n
  • C π‘Ÿ = 2 . 5 1 βˆ’ 1 . 5 πœƒ s i n
  • D π‘Ÿ = 2 . 5 1 + 1 . 5 πœƒ c o s

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