Lesson Worksheet: Polar Equation of a Conic Mathematics • 12th Grade

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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.

  • A𝑟=31+0.5𝜃sin
  • B𝑟=31−0.5𝜃sin
  • C𝑟=31+0.5𝜃cos
  • D𝑟=31−0.5𝜃cos
  • E𝑟=61+0.5𝜃sin

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.

  • A𝑟=53+2𝜃sin
  • B𝑟=53+2𝜃cos
  • C𝑟=53−2𝜃sin
  • D𝑟=53+2𝜃cos

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.

  • A𝑟=𝑒𝑑1+𝑒𝜃cos
  • B𝑟=𝑒𝑑1−𝑒𝜃cos
  • C𝑟=𝑒𝑑1−𝑒𝜃sin
  • D𝑟=𝑒𝑑1+𝑒𝜃sin

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

  • A𝑥=−2.5
  • B0
  • C𝑥=−53
  • D𝑥=53

Hence, write the equation of the conic.

  • A𝑟=2.51+1.5𝜃sin
  • B𝑟=2.51+1.5𝜃cos
  • C𝑟=2.51−1.5𝜃sin
  • D𝑟=2.51−1.5𝜃cos

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.

  • A𝑥=−𝑑
  • B𝑦=𝑑
  • C𝑥=𝑑
  • D𝑦=−𝑑
  • E𝑥=𝑒𝑑

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?

  • A𝑥=−3
  • B𝑦=3
  • C𝑥=−2
  • D𝑥=3
  • E𝑦=−3

Q6:

What are the possible values of the eccentricity 𝑒 in the polar equations of conics?

  • A(1,∞)
  • B[0,∞)
  • C[1,∞)
  • D(−∞,0]
  • E[0,1]

Q7:

A conic with focus at the pole has eccentricity 𝑒=1 and vertex at (−1,0). Write the equation of the conic in polar form.

  • A𝑟=21−(𝜃)sin
  • B𝑟=11−(𝜃)cos
  • C𝑟=21−(𝜃)cos
  • D𝑟=21+(𝜃)sin
  • E𝑟=21+(𝜃)cos

Q8:

State the type of conic that is described by the equation 𝑟=12−(𝜃)sin.

  • AHyperbola
  • BParabola
  • CCircle
  • DEllipse

Q9:

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

  • A𝑥=−𝑑
  • B𝑦=𝑑
  • C𝑦=𝑒𝑑
  • D𝑥=𝑑
  • E𝑦=−𝑑

Q10:

Given the following graph of a conic whose focus is at the pole, find its polar equation.

  • A𝑟=32−(𝜃)cos
  • B𝑟=32+(𝜃)cos
  • C𝑟=32+(𝜃)cos
  • D𝑟=32−(𝜃)sin
  • E𝑟=32+(𝜃)sin

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