Worksheet: Conversion between Rectangular and Polar Equations

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In this worksheet, we will practice converting equations from polar to rectangular form and vice versa.

Q1:

Consider the polar equation 𝑟=2𝜃cos. Complete the following steps to help you find the Cartesian form of the equation by writing an equivalent equation each time.

Multiply both sides of the equation through by 𝑟.

  • A 𝑟 = 𝑟 𝜃 c o s
  • B 𝑟 = 2 𝑟 𝜃 c o s
  • C 𝑟 = 2 𝑟 𝜃 c o s
  • D 𝑟 = 2 𝜃 c o s
  • E 𝑟 = 𝑟 𝜃 c o s

Use the fact that 𝑥=𝑟𝜃cos to simplify the expression.

  • A 𝑟 = 2 𝑥
  • B 𝑟 = 𝑥
  • C 𝑟 = 𝑥
  • D 2 𝑟 = 𝑥
  • E 𝑟 = 2 𝑥

Given that 𝑥=𝑟𝜃cos and 𝑦=𝑟𝜃sin, we can use the Pythagorean theorem to show that 𝑥+𝑦=𝑟. Use this to eliminate the 𝑟 in the previous expression.

  • A 𝑥 + 𝑦 = 2 𝑥
  • B 𝑥 + 𝑦 = 𝑥
  • C 𝑥 + 𝑦 = 𝑥
  • D 𝑥 + 𝑦 = 4 𝑥
  • E 𝑥 + 𝑦 = 𝑥 2

Q2:

Convert 𝑟=2𝜃sec into cartesian form.

  • A 𝑦 = 2
  • B 𝑥 = 2
  • C 𝑥 = 4
  • D 𝑥 = 2
  • E 𝑦 = 2

Q3:

Consider the Cartesian equation 𝑦=2𝑥+3.Complete the following steps to find the polar form of the equation by writing an equivalent equation each time.

First, use the fact that 𝑥=𝑟𝜃cos to eliminate 𝑥.

  • A 𝑦 = 𝑟 𝜃 + 3 c o s
  • B 𝑦 = 2 ( 𝑟 𝜃 + 3 ) c o s
  • C 𝑦 = 2 𝑟 𝜃 c o s
  • D 𝑦 = 2 𝑟 𝜃 − 3 c o s
  • E 𝑦 = 2 𝑟 𝜃 + 3 c o s

Now, use the fact that 𝑦=𝑟𝜃sin to eliminate 𝑦.

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

Finally, make 𝑟 the subject.

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

Q4:

Convert the equation 𝑥+𝑦=25 into polar form.

  • A 𝑟 = 5
  • B 𝑟 = 5 0
  • C 𝑟 = 6 2 5
  • D 𝑟 = 2 5
  • E 𝑟 = 2 5 2

Q5:

Convert the rectangular equation 𝑥+𝑦=25 to the polar form.

  • A 𝑟 = 5
  • B 𝑟 = 5
  • C 𝑟 = √ 5
  • D 𝑟 = 2 5

Q6:

Convert the polar equation 𝑟=4𝜃−6𝜃cossin to the rectangular form.

  • A ( 𝑥 − 2 ) + ( 𝑦 + 3 ) = 1 3
  • B ( 𝑥 − 2 ) − ( 𝑦 + 3 ) = √ 1 3
  • C ( 𝑥 − 2 ) − ( 𝑦 + 3 ) = 1 3
  • D ( 𝑥 + 2 ) + ( 𝑦 − 3 ) = 1 3
  • E ( 𝑥 − 2 ) + ( 𝑦 + 3 ) = √ 1 3

Q7:

Convert the rectangular equation 𝑦=4 to the polar form.

  • A 𝑟 = 2
  • B 𝑟 = 4 𝜃 s e c
  • C 𝑟 = 4 𝜃 c s c
  • D 𝑟 = 4
  • E 𝑟 = 2 𝜃 s e c

Q8:

Convert the polar equation 𝜃=𝜋4 to the rectangular form.

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

Q9:

Convert the polar equation 𝑟=2 to the rectangular form.

  • A 𝑥 + 𝑦 = 2
  • B 𝑥 + 𝑦 = 4
  • C 𝑥 − 𝑦 = 4
  • D 𝑥 − 𝑦 = 2
  • E 𝑥 + 𝑦 = 2

Q10:

Consider the rectangular equation 𝑥−𝑦=25.

Convert the given equation to the polar form.

  • A 𝑟 = 2 5
  • B 𝑟 = √ 5
  • C 𝑟 = 2 5 2 𝜃 c s c
  • D 𝑟 = 2 5
  • E 𝑟 = 2 5 2 𝜃 s e c

Which of the following is the sketch of the equation?

  • A
  • B
  • C
  • D
  • E

Q11:

Convert 𝑟=𝜃cot into Cartesian form.

  • A 𝑦 + 𝑥 𝑦 − 1 = 0
  • B 𝑦 + 𝑥 𝑦 − 1 = 0
  • C 𝑦 + 𝑥 𝑦 − 1 = 0
  • D 𝑦 + 𝑥 𝑦 − 1 = 0
  • E 𝑦 + 𝑥 𝑦 + 1 = 0

Q12:

Convert 𝑟=𝜃−4𝜃cossin into Cartesian form.

  • A 𝑥 − 𝑥 + 𝑦 + 4 𝑦 = 0
  • B 𝑥 − 4 𝑥 + 𝑦 + 𝑦 = 0
  • C 𝑥 + 𝑥 + 𝑦 + 4 𝑦 = 0
  • D 2 𝑥 − 𝑥 + 2 𝑦 + 4 𝑦 = 0
  • E 𝑥 − 2 𝑥 + 𝑦 + 8 𝑦 = 0

Q13:

Convert 𝑥=−5 into polar form.

  • A 𝑟 = − 5 𝜃 s e c
  • B 𝑟 = 5 𝜃 c s c
  • C 𝑟 = 2 5 𝜃 s e c
  • D 𝑟 = 5 𝜃 s e c
  • E 𝑟 = − 5 𝜃 c s c

Q14:

Convert (𝑥−1)+(𝑦−4)=1 into polar form.

  • A 𝑟 − 𝑟 ( 4 𝜃 + 𝜃 ) + 1 6 = 0 c o s s i n
  • B 𝑟 − 𝑟 ( 𝜃 + 4 𝜃 ) + 1 6 = 0 c o s s i n
  • C 𝑟 − 2 𝑟 ( 4 𝜃 + 𝜃 ) + 1 6 = 0 c o s s i n
  • D 𝑟 − 2 𝑟 ( 𝜃 + 4 𝜃 ) + 1 6 = 0 c o s s i n
  • E 𝑟 + 2 𝑟 ( 𝜃 + 4 𝜃 ) + 1 6 = 0 c o s s i n

Q15:

Convert 𝑟=2𝜃sin into Cartesian form.

  • A ( 𝑥 − 1 ) + ( 𝑦 − 1 ) = 1
  • B ( 𝑥 + 1 ) + 𝑦 = 1
  • C 𝑥 + ( 𝑦 + 1 ) = 1
  • D ( 𝑥 − 1 ) + 𝑦 = 1
  • E 𝑥 + ( 𝑦 − 1 ) = 1

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