Worksheet: Polar Equation of a Conic

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=310.5𝜃sin
  • C𝑟=31+0.5𝜃cos
  • D𝑟=310.5𝜃cos
  • 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=532𝜃sin
  • 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=𝑒𝑑1𝑒𝜃cos
  • C𝑟=𝑒𝑑1𝑒𝜃sin
  • 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=2.511.5𝜃sin
  • D𝑟=2.511.5𝜃cos

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=612(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

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