Polar Coordinates Quiz

Grade 12 · Math · Quiz

10 questions on polar coordinates, conversions, and common polar curves.

Questions

1. Convert (2, π/2) from polar to Cartesian.

   1. (0, 2)
   2. (2, 0)
   3. (1, √3)
   4. (√2, √2)
   Show answer

   Answer: (0, 2)

   x = 2 cos(π/2) = 0; y = 2 sin(π/2) = 2.

2. Which polar equation represents a cardioid?

   1. r = 3
   2. r = 3 cos θ
   3. r = 3(1 + sin θ)
   4. r² = 9 cos(2θ)
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   Answer: r = 3(1 + sin θ)

   Cardioids have the form r = a(1 ± sin θ) or r = a(1 ± cos θ).

3. Convert (−4, 0) Cartesian to polar with r > 0 and 0 ≤ θ < 2π.

   Show answer

   r = √(16+0) = 4; θ = arctan(0/−4) but x < 0, so θ = π.

4. A point in polar coordinates has a unique representation.

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   The same point can be represented as (r, θ) = (r, θ + 2π) or even (−r, θ + π).

5. How many petals does r = 2 sin(4θ) have?

   1. 4
   2. 8
   3. 2
   4. 6
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   Answer: 8

   Even n gives 2n petals. n = 4, so 2·4 = 8 petals.

6. Convert r = 6 sin θ to Cartesian form.

   Show answer

   Multiply both sides by r: r² = 6r sin θ → x²+y² = 6y → x² + (y−3)² = 9.

7. What is the maximum r-value of r = 3 + 2 cos θ?

   1. 2
   2. 3
   3. 5
   4. 6
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   Answer: 5

   cos θ is max when θ = 0: r = 3 + 2(1) = 5.

8. The graph of r = −2 is the same as r = 2.

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   r = −2 plots points at distance 2 in the opposite direction, producing the same circle as r = 2.

9. The polar equation of a circle centered at the origin with radius 7 is:

   1. θ = 7
   2. r = 7
   3. r² = 7
   4. r = 7θ
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   Answer: r = 7

   All points equidistant from the pole satisfy r = constant.

10. Find the Cartesian equation of r = sec θ.

    Show answer

    r = 1/cos θ → r cos θ = 1 → x = 1.

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