Worksheet: Conversion between Rectangular and Polar Equations

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.

  • A𝑟=𝑟𝜃cos
  • B𝑟=2𝑟𝜃cos
  • C𝑟=2𝑟𝜃cos
  • D𝑟=2𝜃cos
  • E𝑟=𝑟𝜃cos

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

  • A𝑟=2𝑥
  • B𝑟=𝑥
  • C𝑟=𝑥
  • D2𝑟=𝑥
  • E𝑟=2𝑥

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.

  • A𝑥+𝑦=2𝑥
  • B𝑥+𝑦=𝑥
  • C𝑥+𝑦=𝑥
  • D𝑥+𝑦=4𝑥
  • E𝑥+𝑦=𝑥2

Q2:

Convert 𝑟=2𝜃sec into cartesian form.

  • A𝑦=2
  • B𝑥=2
  • C𝑥=4
  • D𝑥=2
  • E𝑦=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.

  • A𝑦=𝑟𝜃+3cos
  • B𝑦=2(𝑟𝜃+3)cos
  • C𝑦=2𝑟𝜃cos
  • D𝑦=2𝑟𝜃3cos
  • E𝑦=2𝑟𝜃+3cos

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

  • A𝑟𝜃=2(𝑟𝜃+3)sincos
  • B𝑟𝜃=2𝑟𝜃sincos
  • C𝑟𝜃=2𝑟𝜃3sincos
  • D𝑟𝜃=𝑟𝜃+3sincos
  • E𝑟𝜃=2𝑟𝜃+3sincos

Finally, make 𝑟 the subject.

  • A𝑟=3𝜃𝜃sincos
  • B𝑟=3𝜃+𝜃sincos
  • C𝑟=3𝜃+2𝜃sincos
  • D𝑟=3𝜃2𝜃sincos
  • E𝑟=3𝜃2𝜃sincos

Q4:

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

  • A𝑟=5
  • B𝑟=50
  • C𝑟=625
  • D𝑟=25
  • E𝑟=252

Q5:

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

  • A𝑟=5
  • B𝑟=5
  • C𝑟=5
  • D𝑟=25

Q6:

Convert the polar equation 𝑟=4𝜃6𝜃cossin to the rectangular form.

  • A(𝑥2)+(𝑦+3)=13
  • B(𝑥2)(𝑦+3)=13
  • C(𝑥2)(𝑦+3)=13
  • D(𝑥+2)+(𝑦3)=13
  • E(𝑥2)+(𝑦+3)=13

Q7:

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

  • A𝑟=2
  • B𝑟=4𝜃sec
  • C𝑟=4𝜃csc
  • D𝑟=4
  • E𝑟=2𝜃sec

Q8:

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

  • A𝑦=22𝑥
  • B𝑦=22𝑥
  • C𝑦=𝑥
  • D𝑦=22𝑥
  • E𝑦=𝑥

Q9:

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

  • A𝑥+𝑦=2
  • B𝑥+𝑦=4
  • C𝑥𝑦=4
  • D𝑥𝑦=2
  • E𝑥+𝑦=2

Q10:

Consider the rectangular equation 𝑥𝑦=25.

Convert the given equation to the polar form.

  • A𝑟=25
  • B𝑟=5
  • C𝑟=252𝜃csc
  • D𝑟=25
  • E𝑟=252𝜃sec

Which of the following is the sketch of the equation?

  • A
  • B
  • C
  • D
  • E

Q11:

Convert 𝑟=𝜃cot into Cartesian form.

  • A𝑦+𝑥𝑦1=0
  • B𝑦+𝑥𝑦1=0
  • C𝑦+𝑥𝑦1=0
  • D𝑦+𝑥𝑦1=0
  • E𝑦+𝑥𝑦+1=0

Q12:

Convert 𝑟=𝜃4𝜃cossin into Cartesian form.

  • A𝑥𝑥+𝑦+4𝑦=0
  • B𝑥4𝑥+𝑦+𝑦=0
  • C𝑥+𝑥+𝑦+4𝑦=0
  • D2𝑥𝑥+2𝑦+4𝑦=0
  • E𝑥2𝑥+𝑦+8𝑦=0

Q13:

Convert 𝑥=5 into polar form.

  • A𝑟=5𝜃sec
  • B𝑟=5𝜃csc
  • C𝑟=25𝜃sec
  • D𝑟=5𝜃sec
  • E𝑟=5𝜃csc

Q14:

Convert (𝑥1)+(𝑦4)=1 into polar form.

  • A𝑟𝑟(4𝜃+𝜃)+16=0cossin
  • B𝑟𝑟(𝜃+4𝜃)+16=0cossin
  • C𝑟2𝑟(4𝜃+𝜃)+16=0cossin
  • D𝑟2𝑟(𝜃+4𝜃)+16=0cossin
  • E𝑟+2𝑟(𝜃+4𝜃)+16=0cossin

Q15:

Convert 𝑟=2𝜃sin into Cartesian form.

  • A(𝑥1)+(𝑦1)=1
  • B(𝑥+1)+𝑦=1
  • C𝑥+(𝑦+1)=1
  • D(𝑥1)+𝑦=1
  • E𝑥+(𝑦1)=1

Q16:

Convert 𝑦=𝑥 into polar form.

  • A𝜃=𝜋4
  • B𝑟=𝜋4
  • C𝜃=3𝜋4
  • D𝜃=1tan
  • E𝑟𝜃=𝜋4

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