JEE Main - Circle standard and general equations and properties Online Test (MCQ Quiz)

1. 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

2. 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

3. 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)

4. 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)

5. 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

6. 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

7. 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)

8. 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)

9. 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

10. 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

11. 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

12. 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

13. 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)

14. 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

15. 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)

16. The equation x^2 + y^2 = r^2 represents a circle with radius:

A) r^2 B) r C) sqrt(r) D) 0

17. 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)

18. 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)

19. 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|

20. 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|

21. 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

22. 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)

23. 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

24. 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

25. 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

26. 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)

27. 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)

28. 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

29. 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

30. 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

31. 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)

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

Circle standard and general equations and properties - Online Test

Test Results