Polar Coordinates and Straight Lines - One Line Questions

1. If a point has Cartesian coordinates (3, 4), what are its polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π? — (5, arctan(4/3))
2. If a line has polar equation r = 2 cos(θ), what is its Cartesian equation? — (x - 1)² + y² = 1
3. Convert the polar equation r = 2a cos(θ) to Cartesian form. — (x - a)² + y² = a²
4. What is the distance 'p' from the origin for the line r cos(θ - π/4) = √2? — √2
5. What is the angle of the line passing through the origin with polar equation θ = 5π/6? — 150°
6. Consider the line with polar equation r = 4 / (2 cos(θ) - sin(θ)). What is its Cartesian equivalent? — 2x - y = 4
7. What is the angle 'α' in the polar equation of a line r cos(θ - α) = p if the line is perpendicular to the line θ = π/6? — 2π/3
8. If the polar equation of a line is r cos(θ - 2π/3) = 3, what is the distance 'p' from the origin? — 3
9. If the polar equation of a line is r = 6 / (3 cos(θ) + 4 sin(θ)), what is the Cartesian equation? — 3x + 4y = 6
10. The polar equation r = 2 is a circle. What is its center and radius in Cartesian coordinates? — Center (0,0), Radius 2
11. What is the general polar equation of a circle passing through the origin with its center at (a, α) in polar coordinates? — r = 2a cos(θ - α)
12. What is the polar equation of a circle centered at the origin with radius 'a'? — r = a
13. What is the general polar equation of a straight line that does not pass through the origin? — r cos(θ - α) = p
14. What is the polar equation of the line that is 2 units away from the origin and makes an angle of 30° with the polar axis? — r cos(θ - π/6) = 2
15. What is the polar equation of the line x = 3? — r cos(θ) = 3
16. What is the polar form of the line x = c? — r cos(θ) = c
17. What is the polar equation of a line perpendicular to the polar axis at a distance 'd' from the origin? — r cos(θ) = d
18. What is the polar equation of the line passing through (p, α) perpendicular to the initial line? — r cos(θ) = p
19. What is the polar equation of the line y = -2? — r sin(θ) = -2
20. What is the polar equation of a line parallel to the polar axis at a distance 'd' from it? — r sin(θ) = d
21. What is the polar equation of the line passing through (p, α) parallel to the initial line? — r sin(θ) = p
22. The Cartesian equation 2x - y = 5 is equivalent to which polar equation? — r(2 cos(θ) - sin(θ)) = 5
23. What is the polar equation of the line 3x + 4y = 12? — r(3 cos(θ) + 4 sin(θ)) = 12
24. Convert the Cartesian equation Ax + By + C = 0 to its polar form. — r(A cos(θ) + B sin(θ)) = -C
25. What is the polar equation of the line passing through (r₁, θ₁) and (r₂, θ₂)? — r(sin(θ₂ - θ)cos(θ₁) - sin(θ₁ - θ)cos(θ₂)) = r₁r₂sin(θ₂ - θ₁)
26. What is the polar equation of the line passing through (2, π/6) and (4, π/3)? — r(sin(π/3 - θ)cos(π/6) - sin(π/6 - θ)cos(π/3)) = 8sin(π/3 - π/6)
27. What is the polar form of the line passing through the origin with slope m in Cartesian coordinates? — tan(θ) = m
28. In the polar equation of a line r cos(θ - α) = p, what does 'α' represent? — The angle the normal to the line makes with the polar axis
29. In the polar equation of a line r cos(θ - α) = p, what does 'p' represent? — The perpendicular distance from the origin to the line
30. The polar equation r = 2 / (cos(θ) + sin(θ)) represents which Cartesian line? — x + y = 2
31. Consider the line with polar equation r cos(θ - π) = 5. What is its Cartesian equivalent? — x = -5
32. The polar equation r = sec(θ) represents which Cartesian line? — x = 1
33. Consider the line with polar equation r cos(θ) = 4. What is its Cartesian equivalent? — x = 4
34. What is the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ)? — x = r cos(θ), y = r sin(θ)
35. Convert the polar equation r = 2b sin(θ) to Cartesian form. — x² + (y - b)² = b²
36. If a line has polar equation r = -3 sin(θ), what is its Cartesian equation? — x² + (y + 3/2)² = 9/4
37. Convert the polar equation r = 5 to its Cartesian equivalent. — x² + y² = 25
38. Consider the line with polar equation r sin(θ) = -5. What is its Cartesian equivalent? — y = -5
39. Convert the line θ = 3π/4 to Cartesian form. — y = -x
40. The polar equation r = 3 cos(θ - π/2) represents which Cartesian line? — y = 0
41. The polar equation r = csc(θ) represents which Cartesian line? — y = 1
42. If a line has polar equation r = 2 / sin(θ), what is its Cartesian equation? — y = 2
43. Convert the polar equation θ = π/4 to its Cartesian equivalent. — y = x
44. What is the polar equation of the line passing through the origin with an inclination of 135°? — θ = 3π/4
45. What is the polar equation of the line x + y = 0? — θ = 3π/4 or θ = 7π/4
46. What is the polar equation of the line x = -y? — θ = 3π/4 or θ = 7π/4
47. If a line passes through the pole (origin), what is its polar equation? — θ = constant
48. What is the polar equation of the line passing through (0, 0) and making an angle of 60° with the positive x-axis? — θ = π/3
49. If the polar equation of a line is r cos(θ - π/3) = 2, what is the angle 'α' in the form r cos(θ - α) = p? — π/3
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