Worksheet: Polar Equation of a Conic

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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 , − 6 ) and ( 0 , 2 ) . Write the equation of the conic in polar form.

  • A 𝑟 = 3 1 − 0 . 5 𝜃 c o s
  • B 𝑟 = 3 1 − 0 . 5 𝜃 s i n
  • C 𝑟 = 3 1 + 0 . 5 𝜃 c o s
  • D 𝑟 = 3 1 + 0 . 5 𝜃 s i n
  • E 𝑟 = 6 1 + 0 . 5 𝜃 s i n

Q2:

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

Identify the type of the conic.

  • AParabola
  • BCircle
  • CHyperbola
  • DEllipse

Write the equation of the conic in polar form.

  • A 𝑟 = 5 3 + 2 𝜃 c o s
  • B 𝑟 = 5 3 + 2 𝜃 c o s
  • C 𝑟 = 5 3 + 2 𝜃 s i n
  • D 𝑟 = 5 3 − 2 𝜃 s i n

Q3:

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

Identify the type of the conic.

  • AEllipse
  • BParabola
  • CCircle
  • DHyperbola

By deciding whether the directrix is in the form 𝑥 = 𝑑 , 𝑥 = − 𝑑 , 𝑦 = 𝑑 , or 𝑦 = − 𝑑 , where 𝑑 > 0 , select the form of the polar equation of the conic.

  • A 𝑟 = 𝑒 𝑑 1 + 𝑒 𝜃 s i n
  • B 𝑟 = 𝑒 𝑑 1 − 𝑒 𝜃 s i n
  • C 𝑟 = 𝑒 𝑑 1 + 𝑒 𝜃 c o s
  • D 𝑟 = 𝑒 𝑑 1 − 𝑒 𝜃 c o s

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

  • A 𝑥 = − 5 3
  • B 𝑥 = − 2 . 5
  • C0
  • D 𝑥 = 5 3

Hence, write the equation of the conic.

  • A 𝑟 = 2 . 5 1 − 1 . 5 𝜃 c o s
  • B 𝑟 = 2 . 5 1 + 1 . 5 𝜃 s i n
  • C 𝑟 = 2 . 5 1 − 1 . 5 𝜃 s i n
  • D 𝑟 = 2 . 5 1 + 1 . 5 𝜃 c o s

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