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

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

  • Aπ‘Ÿ=31+0.5πœƒsin
  • Bπ‘Ÿ=31βˆ’0.5πœƒsin
  • Cπ‘Ÿ=31+0.5πœƒcos
  • Dπ‘Ÿ=31βˆ’0.5πœƒcos
  • Eπ‘Ÿ=61+0.5πœƒsin

Q2:

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

Identify the type of the conic.

  • AEllipse
  • BHyperbola
  • CParabola
  • DCircle

Write the equation of the conic in polar form.

  • Aπ‘Ÿ=53+2πœƒsin
  • Bπ‘Ÿ=53+2πœƒcos
  • Cπ‘Ÿ=53βˆ’2πœƒsin
  • Dπ‘Ÿ=53+2πœƒcos

Q3:

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

Identify the type of the conic.

  • ACircle
  • BEllipse
  • CHyperbola
  • DParabola

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

  • Aπ‘Ÿ=𝑒𝑑1+π‘’πœƒcos
  • Bπ‘Ÿ=𝑒𝑑1βˆ’π‘’πœƒcos
  • Cπ‘Ÿ=𝑒𝑑1βˆ’π‘’πœƒsin
  • Dπ‘Ÿ=𝑒𝑑1+π‘’πœƒsin

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

  • Aπ‘₯=βˆ’2.5
  • B0
  • Cπ‘₯=βˆ’53
  • Dπ‘₯=53

Hence, write the equation of the conic.

  • Aπ‘Ÿ=2.51+1.5πœƒsin
  • Bπ‘Ÿ=2.51+1.5πœƒcos
  • Cπ‘Ÿ=2.51βˆ’1.5πœƒsin
  • Dπ‘Ÿ=2.51βˆ’1.5πœƒcos

Q4:

Consider the polar equation π‘Ÿ=𝑒𝑑1+𝑒(πœƒ)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.

  • Aπ‘₯=βˆ’π‘‘
  • B𝑦=𝑑
  • Cπ‘₯=𝑑
  • D𝑦=βˆ’π‘‘
  • Eπ‘₯=𝑒𝑑

Q5:

Consider the following polar equation of a conic: π‘Ÿ=61βˆ’2(πœƒ)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?

  • Aπ‘₯=βˆ’3
  • B𝑦=3
  • Cπ‘₯=βˆ’2
  • Dπ‘₯=3
  • E𝑦=βˆ’3

Q6:

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

  • A(1,∞)
  • B[0,∞)
  • C[1,∞)
  • D(βˆ’βˆž,0]
  • E[0,1]

Q7:

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

  • Aπ‘Ÿ=21βˆ’(πœƒ)sin
  • Bπ‘Ÿ=11βˆ’(πœƒ)cos
  • Cπ‘Ÿ=21βˆ’(πœƒ)cos
  • Dπ‘Ÿ=21+(πœƒ)sin
  • Eπ‘Ÿ=21+(πœƒ)cos

Q8:

State the type of conic that is described by the equation π‘Ÿ=12βˆ’(πœƒ)sin.

  • AHyperbola
  • BParabola
  • CCircle
  • DEllipse

Q9:

Given the polar equation π‘Ÿ=𝑒𝑑1+𝑒(πœƒ)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.

  • Aπ‘₯=βˆ’π‘‘
  • B𝑦=𝑑
  • C𝑦=𝑒𝑑
  • Dπ‘₯=𝑑
  • E𝑦=βˆ’π‘‘

Q10:

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

  • Aπ‘Ÿ=32βˆ’(πœƒ)cos
  • Bπ‘Ÿ=32+(πœƒ)cos
  • Cπ‘Ÿ=32+(πœƒ)cos
  • Dπ‘Ÿ=32βˆ’(πœƒ)sin
  • Eπ‘Ÿ=32+(πœƒ)sin

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