Circle standard and general equations and properties - Question Bank

1. The equation x^2 + y^2 = 0 represents:
A) A point circle at the origin
B) An imaginary circle
C) A circle with infinite radius
D) A line
2. What is the equation of the circle with center (a cos θ, a sin θ) and radius a?
A) (x - a cos θ)^2 + (y - a sin θ)^2 = a^2
B) x^2 + y^2 = a^2
C) (x - a cos θ)^2 + (y - a sin θ)^2 = a
D) (x + a cos θ)^2 + (y + a sin θ)^2 = a^2
3. Find the equation of the circle passing through (1, 2) and having its center at the intersection of 2x + y = 3 and x - y = 0.
A) (x - 1)^2 + (y - 1)^2 = 1
B) (x - 1)^2 + (y - 1)^2 = 2
C) (x - 1)^2 + (y - 1)^2 = 3
D) (x - 1)^2 + (y - 1)^2 = 0
4. If the power of a point P with respect to a circle is positive, then P lies:
A) Inside the circle
B) Outside the circle
C) On the circle
D) At the center of the circle
5. If the power of a point P with respect to a circle is negative, then P lies:
A) Inside the circle
B) Outside the circle
C) On the circle
D) At the center of the circle
6. If the power of a point P with respect to a circle is zero, then P lies:
A) Inside the circle
B) Outside the circle
C) On the circle
D) At the center of the circle
7. What is the power of the point (x1, y1) with respect to the circle S = x^2 + y^2 + 2gx + 2fy + c = 0?
A) x1^2 + y1^2 + 2gx1 + 2fy1 + c
B) sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c)
C) (x1 - g)^2 + (y1 - f)^2 - (g^2 + f^2 - c)
D) g^2 + f^2 - c
8. The equation x^2 + y^2 = a^2 is the standard equation of a circle with:
A) Center (a, 0) and radius a
B) Center (0, 0) and radius a
C) Center (0, 0) and radius a^2
D) Center (a, a) and radius a
9. Find the equation of the circle passing through the points (0,0), (a,0), and (0,b).
A) x^2 + y^2 - ax - by = 0
B) x^2 + y^2 + ax + by = 0
C) x^2 + y^2 - ax + by = 0
D) x^2 + y^2 + ax - by = 0
10. What is the equation of the circle whose center is (5, -4) and circumference is 12π?
A) (x - 5)^2 + (y + 4)^2 = 36
B) (x - 5)^2 + (y + 4)^2 = 9
C) (x + 5)^2 + (y - 4)^2 = 36
D) (x - 5)^2 + (y + 4)^2 = 144
11. The equation x^2 + y^2 + 2x - 4y + k = 0 represents a circle. If the radius is 3, find the value of k.
A) -6
B) 6
C) 4
D) -4
12. What is the equation of the circle passing through (a, b) and concentric with x^2 + y^2 + 2gx + 2fy + c = 0?
A) (x - g)^2 + (y - f)^2 = (a - g)^2 + (b - f)^2
B) x^2 + y^2 + 2gx + 2fy + c = (a - g)^2 + (b - f)^2
C) x^2 + y^2 = a^2 + b^2
D) (x + g)^2 + (y + f)^2 = (a + g)^2 + (b + f)^2
13. Find the equation of the circle which touches both axes and has its center in the first quadrant and radius 5.
A) (x - 5)^2 + (y - 5)^2 = 25
B) (x + 5)^2 + (y + 5)^2 = 25
C) (x - 5)^2 + (y + 5)^2 = 25
D) (x + 5)^2 + (y - 5)^2 = 25
14. What is the equation of the circle whose center lies on the line y = x and passes through (2, 0) and (0, 4)?
A) (x - 2)^2 + (y - 2)^2 = 4
B) (x - 2)^2 + (y - 2)^2 = 8
C) (x - 4)^2 + (y - 4)^2 = 16
D) (x + 2)^2 + (y + 2)^2 = 8
15. The equation x^2 + y^2 + 2gx + 2fy + c = 0 represents a circle. What is the condition for it to pass through the origin?
A) c = 0
B) g = 0
C) f = 0
D) g + f + c = 0
16. Find the equation of the circle with center (3, -2) and touching the line 3x - 4y + 5 = 0.
A) (x - 3)^2 + (y + 2)^2 = (26/5)^2
B) (x - 3)^2 + (y + 2)^2 = (17/5)^2
C) (x - 3)^2 + (y + 2)^2 = (22/5)^2
D) (x - 3)^2 + (y + 2)^2 = (20/5)^2
17. What is the equation of the circle passing through the origin and cutting the axes at intercepts a and b on x-axis and y-axis respectively?
A) x^2 + y^2 - ax - by = 0
B) x^2 + y^2 + ax + by = 0
C) x^2 + y^2 - ax + by = 0
D) x^2 + y^2 + ax - by = 0
18. If the two circles x^2 + y^2 + 2g1x + 2f1y + c1 = 0 and x^2 + y^2 + 2g2x + 2f2y + c2 = 0 are concentric, then:
A) g1 = g2, f1 = f2
B) g1 = -g2, f1 = -f2
C) g1 = c1, f1 = c2
D) g1 = f1, g2 = f2
19. The equation x^2 + y^2 + 2gx + 2fy + c = 0 represents a circle of infinite radius when:
A) g^2 + f^2 - c = 0
B) g^2 + f^2 - c > 0
C) A = 0 (in Ax^2 + Ay^2...)
D) g = 0, f = 0, c = 0
20. What is the equation of the circle with center (1, 2) passing through the point (4, 6)?
A) (x - 1)^2 + (y - 2)^2 = 5
B) (x - 1)^2 + (y - 2)^2 = 25
C) (x - 4)^2 + (y - 6)^2 = 5
D) (x - 1)^2 + (y - 2)^2 = sqrt(5)
21. The equation (x-a)^2 + (y-b)^2 = 0 represents:
A) A circle with center (a,b) and radius 0
B) A point circle at the origin
C) An imaginary circle
D) A line segment
22. What is the locus of a point that moves such that its distance from a fixed point is constant?
A) A straight line
B) A circle
C) A parabola
D) An ellipse
23. Find the equation of the circle passing through the points (1, 1), (1, -1), and (-1, 1).
A) x^2 + y^2 = 1
B) x^2 + y^2 - x - y = 0
C) x^2 + y^2 - 2x = 0
D) x^2 + y^2 - 2y = 0
24. What is the radius of the circle represented by Ax^2 + Ay^2 + 2Gx + 2Fy + C = 0?
A) sqrt(G^2/A^2 + F^2/A^2 - C/A)
B) sqrt(G^2 + F^2 - C)
C) sqrt(A(G^2 + F^2) - C)
D) sqrt(G^2 + F^2 + C/A)
25. What is the center of the circle represented by Ax^2 + Ay^2 + 2Gx + 2Fy + C = 0?
A) (-G/A, -F/A)
B) (G/A, F/A)
C) (-G, -F)
D) (G, F)
26. The equation Ax^2 + Ay^2 + 2Gx + 2Fy + C = 0 represents a circle if:
A) G^2 + F^2 - AC > 0
B) G^2 + F^2 - AC < 0
C) A(G^2 + F^2) - C > 0
D) A(G^2 + F^2) - C < 0
27. What is the equation of the circle passing through (0,0) and concentric with x^2 + y^2 - 4x + 6y - 12 = 0?
A) x^2 + y^2 - 4x + 6y = 0
B) x^2 + y^2 = 13
C) x^2 + y^2 = 12
D) x^2 + y^2 + 4x - 6y = 0
28. The equation 2x^2 + 2y^2 - 3x + 5y - 7 = 0 represents a circle. What is its radius?
A) sqrt(3^2 + 5^2 - 4*(-7))/4
B) sqrt(3^2 + 5^2 + 4*(-7))/4
C) sqrt(9 + 25 + 28)/4
D) sqrt(62)/4
29. The equation 2x^2 + 2y^2 - 3x + 5y - 7 = 0 represents a circle. What is its center?
A) (3/4, -5/4)
B) (-3/4, 5/4)
C) (3/2, -5/2)
D) (-3/2, 5/2)
30. What is the equation of the circle whose center is at the origin and radius is R?
A) x^2 - y^2 = R^2
B) x^2 + y^2 = R^2
C) (x-R)^2 + (y-R)^2 = R^2
D) x^2 + y^2 = R
31. Find the equation of the circle with center (h, k) touching the y-axis.
A) (x - h)^2 + (y - k)^2 = h^2
B) (x - h)^2 + (y - k)^2 = k^2
C) (x - h)^2 + (y - k)^2 = r^2
D) (x - h)^2 + (y - k)^2 = |h|
32. Find the equation of the circle with center (h, k) touching the x-axis.
A) (x - h)^2 + (y - k)^2 = k^2
B) (x - h)^2 + (y - k)^2 = h^2
C) (x - h)^2 + (y - k)^2 = r^2
D) (x - h)^2 + (y - k)^2 = |k|
33. What is the length of the tangent from an external point (x1, y1) to the circle x^2 + y^2 + 2gx + 2fy + c = 0?
A) sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c)
B) x1^2 + y1^2 + 2gx1 + 2fy1 + c
C) sqrt(x1^2 + y1^2 - 2gx1 - 2fy1 - c)
D) sqrt(g^2 + f^2 - c)
34. What is the condition for the line y = mx + c to be a tangent to the circle x^2 + y^2 = a^2?
A) c^2 = a^2(m^2 + 1)
B) c^2 = a^2(m^2 - 1)
C) c^2 = a^2(1 - m^2)
D) c = a^2(m^2 + 1)
35. The equation x^2 + y^2 = r^2 represents a circle with radius:
A) r^2
B) r
C) sqrt(r)
D) 0
36. The equation of a circle is x^2 + y^2 = r^2. What is its center?
A) (r, 0)
B) (0, r)
C) (0, 0)
D) (r, r)
37. What is the radius of the circle passing through the points (0,0), (a,0), and (0,b)?
A) sqrt(a^2 + b^2)
B) sqrt(a^2 + b^2)/2
C) sqrt(a^2 - b^2)/2
D) sqrt(b^2 - a^2)/2
38. What is the center of the circle passing through the points (0,0), (a,0), and (0,b)?
A) (a/2, b/2)
B) (a, b)
C) (a/2, b)
D) (a, b/2)
39. Find the equation of the circle whose diameter has endpoints (1, 2) and (4, 5).
A) (x - 5/2)^2 + (y - 7/2)^2 = 25/4
B) (x - 5/2)^2 + (y - 7/2)^2 = 45/4
C) (x - 1)^2 + (y - 2)^2 = 13
D) (x - 4)^2 + (y - 5)^2 = 13
40. If g^2 + f^2 - c < 0 for the general equation of a circle, what does the equation represent?
A) A point circle
B) An imaginary circle
C) A real circle
D) A hyperbola
41. If g^2 + f^2 - c = 0 for the general equation of a circle, what does the equation represent?
A) A point circle
B) An imaginary circle
C) A straight line
D) A parabola
42. The equation x^2 + y^2 + 2gx + 2fy + c = 0 represents a circle if:
A) g^2 + f^2 - c > 0
B) g^2 + f^2 - c < 0
C) g^2 + f^2 + c > 0
D) g^2 + f^2 + c < 0
43. What is the radius of the circle given by the equation x^2 + y^2 - 4x + 6y - 12 = 0?
A) sqrt(12)
B) sqrt(25)
C) sqrt(17)
D) sqrt(13)
44. What is the center of the circle given by the equation x^2 + y^2 - 4x + 6y - 12 = 0?
A) (2, -3)
B) (-2, 3)
C) (4, -6)
D) (-4, 6)
45. Find the equation of the circle with center (2, -3) and radius 4.
A) (x - 2)^2 + (y + 3)^2 = 16
B) (x + 2)^2 + (y - 3)^2 = 16
C) (x - 2)^2 + (y + 3)^2 = 4
D) (x + 2)^2 + (y - 3)^2 = 4
46. A circle passes through the origin (0,0) and has its center at (3,4). What is its equation?
A) (x - 3)^2 + (y - 4)^2 = 5
B) (x - 3)^2 + (y - 4)^2 = 25
C) (x + 3)^2 + (y + 4)^2 = 5
D) (x + 3)^2 + (y + 4)^2 = 25
47. For the general equation of a circle x^2 + y^2 + 2gx + 2fy + c = 0, what is the radius?
A) sqrt(g^2 + f^2 - c)
B) sqrt(g^2 + f^2 + c)
C) g^2 + f^2 - c
D) sqrt(g^2 - f^2 - c)
48. For the general equation of a circle x^2 + y^2 + 2gx + 2fy + c = 0, what are the coordinates of the center?
A) (g, f)
B) (-g, -f)
C) (g, -f)
D) (-g, f)
49. What is the general equation of a circle?
A) x^2 + y^2 + 2gx + 2fy + c = 0
B) x^2 - y^2 + 2gx + 2fy + c = 0
C) x^2 + y^2 - 2gx - 2fy + c = 0
D) x^2 + y^2 + gx + fy + c = 0
50. What is the standard equation of a circle with center (h, k) and radius r?
A) (x + h)^2 + (y + k)^2 = r^2
B) (x - h)^2 + (y - k)^2 = r^2
C) (x - h)^2 - (y - k)^2 = r^2
D) (x + h)^2 - (y + k)^2 = r^2
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