Worksheet: Polar Equation of a Conic

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In this worksheet, we will practice determining the type of a conic section (ellipse, parabola, or hyperbola) and writing polar equations of conics given the eccentricity and some other characteristic.

Q1:

A conic with focus at the pole has eccentricity 𝑒=12 and vertices at (0,−6) and (0,2). Write the equation of the conic in polar form.

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

Q2:

A conic with focus at the pole has eccentricity 𝑒=23 and directrix 𝑦=−52.

Identify the type of the conic.

  • AEllipse
  • BHyperbola
  • CParabola
  • DCircle

Write the equation of the conic in polar form.

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

Q3:

A conic with its focus at the pole has eccentricity 𝑒=32 and vertices at (−1,0) and (−5,0).

Identify the type of the conic.

  • ACircle
  • BEllipse
  • CHyperbola
  • DParabola

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

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

  • A 𝑥 = − 2 . 5
  • B0
  • C 𝑥 = − 5 3
  • D 𝑥 = 5 3

Hence, write the equation of the conic.

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

Q4:

Consider the polar equation 𝑟=𝑒𝑑1+𝑒(𝜃)cos of a conic with its focus at the pole and eccentricity 𝑒, where 𝑒>0 and 𝑑>0.

State the equation of the directrix.

  • A 𝑥 = − 𝑑
  • B 𝑦 = 𝑑
  • C 𝑥 = 𝑑
  • D 𝑦 = − 𝑑
  • E 𝑥 = 𝑒 𝑑

Q5:

Consider the following polar equation of a conic: 𝑟=61−2(𝜃)cos.

Determine the value of the eccentricity.

State the type of conic that is described by the equation.

  • AEllipse
  • BParabola
  • CHyperbola
  • DCircle

Which of the following is a directrix of the conic?

  • A 𝑥 = − 3
  • B 𝑦 = 3
  • C 𝑥 = − 2
  • D 𝑥 = 3
  • E 𝑦 = − 3

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