Locus and its equation - Question Bank

1. What is the locus of a point P(x, y) that moves such that its distance from the origin is equal to its distance from the line x = 5?
A) A parabola with vertex (2.5, 0)
B) A parabola with vertex (0, 2.5)
C) A circle with center (2.5, 0)
D) An ellipse with foci (0, 0) and (5, 0)
2. Find the equation of the locus of a point P(x, y) that is equidistant from the point (a, 0) and the y-axis.
A) y^2 = 4ax - a^2
B) y^2 = -4ax + a^2
C) x^2 = 4ay - a^2
D) x^2 = -4ay + a^2
3. The locus of a point P(x, y) such that the distance from P to A(1, 0) is equal to its distance from the line x = 5, is:
A) y^2 = -8(x - 3)
B) y^2 = 8(x - 3)
C) x^2 = -8(y - 3)
D) x^2 = 8(y - 3)
4. What is the locus of a point P(x, y) that moves such that the sum of its distances from (0, 3) and (0, -3) is 10?
A) x^2/16 + y^2/25 = 1
B) x^2/25 + y^2/16 = 1
C) x^2/9 + y^2/10 = 1
D) x^2/10 + y^2/9 = 1
5. Find the equation of the locus of a point P(x, y) that is equidistant from the point (0, 0) and the line x = 4.
A) y^2 = -8(x - 2)
B) y^2 = 8(x - 2)
C) x^2 = -8(y - 2)
D) x^2 = 8(y - 2)
6. The locus of a point P(x, y) such that the difference of its distances from the points (2, 0) and (-2, 0) is 2, is:
A) x^2/1 - y^2/3 = 1
B) x^2/3 - y^2/1 = 1
C) x^2/1 + y^2/3 = 1
D) x^2/3 + y^2/1 = 1
7. What is the locus of a point P(x, y) such that the distance from P to the point (0, 2) is equal to the distance from P to the line y = -2?
A) x^2 = 8y
B) x^2 = -8y
C) y^2 = 8x
D) y^2 = -8x
8. Find the equation of the locus of a point P(x, y) that moves such that the square of its distance from the origin is equal to its distance from the x-axis.
A) x^2 + y^2 = |y|
B) x^2 + y^2 = |x|
C) x^2 + y^2 = y
D) x^2 + y^2 = x
9. The locus of a point P(x, y) such that the distance from P to A(2, 0) is equal to its distance from the y-axis is:
A) y^2 = 4x - 4
B) y^2 = -4x + 4
C) x^2 = 4y - 4
D) x^2 = -4y + 4
10. What is the locus of a point P(x, y) such that the sum of its distances from (3, 0) and (-3, 0) is 10?
A) x^2/25 + y^2/16 = 1
B) x^2/16 + y^2/25 = 1
C) x^2/9 + y^2/10 = 1
D) x^2/10 + y^2/9 = 1
11. Find the equation of the locus of a point P(x, y) that is equidistant from the y-axis and the point (0, 4).
A) x^2 = -8(y - 2)
B) x^2 = 8(y - 2)
C) y^2 = -8(x - 2)
D) y^2 = 8(x - 2)
12. The locus of a point P(x, y) such that the distance from P to (2, 3) is equal to the distance from P to the line y = -1, is:
A) A parabola
B) An ellipse
C) A hyperbola
D) A circle
13. Find the equation of the locus of a point P(x, y) that moves such that its distance from the point (0, 0) is equal to its distance from the line 4x + 3y - 12 = 0.
A) (4x + 3y - 12)^2 = 25(x^2 + y^2)
B) (4x + 3y - 12)^2 = 5(x^2 + y^2)
C) 25(4x + 3y - 12)^2 = x^2 + y^2
D) 5(4x + 3y - 12)^2 = x^2 + y^2
14. What is the locus of a point P(x, y) such that the distance from P to the origin is equal to the distance from P to the line y = -4?
A) A parabola x^2 = 8(y + 2)
B) A parabola x^2 = -8(y + 2)
C) A parabola y^2 = 8(x + 2)
D) A parabola y^2 = -8(x + 2)
15. The locus of a point P(x, y) such that the sum of its distances from (a, 0) and (-a, 0) is 2c (where c > a), is:
A) x^2/c^2 + y^2/(c^2 - a^2) = 1
B) x^2/(c^2 - a^2) + y^2/c^2 = 1
C) x^2/a^2 + y^2/c^2 = 1
D) x^2/c^2 + y^2/a^2 = 1
16. Find the equation of the locus of a point P(x, y) that is equidistant from the point (2, 3) and the line x = 5.
A) y^2 - 6y + 13 = (x - 5)^2
B) x^2 - 10x + 25 = (y - 3)^2
C) y^2 - 6y + 9 = (x - 2)^2
D) y^2 - 6y + 9 = (5 - x)^2
17. What is the locus of a point P(x, y) that moves such that the line segment joining P to the origin is perpendicular to the line segment joining P to the point (5, 0)?
A) A circle with diameter from (0,0) to (5,0)
B) A circle with center (2.5, 0) and radius 2.5
C) A parabola
D) An ellipse
18. The locus of a point P(x, y) such that the difference of its distances from the points (0, 2) and (0, -2) is 2, is:
A) x^2/1 + y^2/3 = 1
B) x^2/3 + y^2/1 = 1
C) x^2/1 - y^2/3 = 1
D) x^2/3 - y^2/1 = 1
19. Find the equation of the locus of a point P(x, y) such that the distance from P to A(1, 0) is equal to its distance from the line x = -1.
A) y^2 = 4x
B) y^2 = -4x
C) x^2 = 4y
D) x^2 = -4y
20. What is the locus of a point P(x, y) that moves such that its distance from the point (1, 1) is equal to its distance from the line x + y = 0?
A) A parabola
B) An ellipse
C) A hyperbola
D) A circle
21. The locus of a point P(x, y) such that the sum of its distances from (0, 2) and (0, -2) is 8 is:
A) x^2/12 + y^2/16 = 1
B) x^2/16 + y^2/12 = 1
C) x^2/4 + y^2/16 = 1
D) x^2/16 + y^2/4 = 1
22. Find the equation of the locus of a point P(x, y) that is equidistant from the point (a, b) and the x-axis.
A) (x - a)^2 + (y - b)^2 = y^2
B) (x - a)^2 + (y - b)^2 = x^2
C) y^2 - 2by + b^2 = (x - a)^2
D) x^2 - 2ax + a^2 = (y - b)^2
23. What is the locus of a point P(x, y) such that the distance from P to the point (0, 0) is equal to the distance from P to the line y = 4?
A) A parabola x^2 = -8(y - 2)
B) A parabola x^2 = 8(y - 2)
C) A parabola y^2 = -8(x - 2)
D) A parabola y^2 = 8(x - 2)
24. Find the equation of the locus of a point P(x, y) that moves such that its distance from the point (1, 2) is twice its distance from the point (3, 4).
A) 3x^2 + 3y^2 + 10x + 16y - 43 = 0
B) 3x^2 + 3y^2 - 10x - 16y + 43 = 0
C) x^2 + y^2 + 10x + 16y - 43 = 0
D) x^2 + y^2 - 10x - 16y + 43 = 0
25. The locus of a point P(x, y) such that the distance from P to the line 3x + 4y + 5 = 0 is equal to the distance from P to the origin, is:
A) A parabola
B) An ellipse
C) A hyperbola
D) A circle
26. What is the locus of a point P(x, y) such that the sum of the squares of its distances from the axes is 16?
A) x^2 + y^2 = 16
B) x^2 + y^2 = 4
C) x^2 + y^2 = 8
D) x^2 + y^2 = 32
27. Find the equation of the locus of a point P(x, y) that is equidistant from the point (3, 0) and the y-axis.
A) (x - 3)^2 + y^2 = x^2
B) (x - 3)^2 + y^2 = y^2
C) x^2 + y^2 = 3
D) y^2 = 6x - 9
28. The locus of a point P(x, y) such that the distance from P to the point (2, 0) is equal to the distance from P to the line x = -2, is:
A) A parabola y^2 = 8x
B) A parabola y^2 = -8x
C) A parabola x^2 = 8y
D) A parabola x^2 = -8y
29. Find the equation of the locus of a point P(x, y) that moves such that the sum of its distances from A(0, 2) and B(0, -2) is 6.
A) x^2/5 + y^2/9 = 1
B) x^2/9 + y^2/5 = 1
C) x^2/5 + y^2/6 = 1
D) x^2/6 + y^2/5 = 1
30. What is the locus of a point P(x, y) such that the distance from P to the origin is equal to the distance from P to the line x = 5?
A) A parabola with vertex at (2.5, 0)
B) A parabola with vertex at (0, 2.5)
C) A circle with center at (2.5, 0)
D) An ellipse with foci at (0, 0) and (5, 0)
31. The locus of a point P(x, y) such that the product of its distances from the points (a, 0) and (-a, 0) is b^2, is:
A) x^2/a^2 + y^2/b^2 = 1
B) x^2/b^2 + y^2/a^2 = 1
C) (x^2 - a^2)/b^2 + y^2/b^2 = 1
D) x^2/a^2 + y^2/(b^2-a^2) = 1
32. Find the equation of the locus of a point P(x, y) that moves such that its distance from the x-axis is half its distance from the y-axis.
A) y^2 = x^2/4
B) x^2 = y^2/4
C) y = x/2
D) x = y/2
33. What is the locus of a point P(x, y) that moves such that its distance from the point (2, 3) is equal to its distance from the line y = 1?
A) A circle
B) A parabola
C) An ellipse
D) A hyperbola
34. The locus of a point P(x, y) such that the sum of the squares of its distances from the points (1, 0) and (-1, 0) is 10, is:
A) x^2 + y^2 = 4
B) x^2 + y^2 = 10
C) x^2 + y^2 = 5
D) x^2 + y^2 = 8
35. Find the equation of the locus of a point P(x, y) which moves such that it is always equidistant from the points (a, 0) and (0, b).
A) bx + ay = (a^2 + b^2)/2
B) ax + by = (a^2 + b^2)/2
C) bx - ay = (a^2 - b^2)/2
D) ax - by = (a^2 - b^2)/2
36. What is the locus of a point P(x, y) such that OP = 2, where O is the origin?
A) x^2 + y^2 = 4
B) x^2 + y^2 = 2
C) x + y = 2
D) x = 2
37. The equation of the locus of a point P(x, y) that moves such that its distance from A(2, 1) is equal to its distance from the y-axis is:
A) (x - 2)^2 + (y - 1)^2 = x^2
B) (x - 2)^2 + (y - 1)^2 = y^2
C) x^2 + y^2 = x
D) x^2 + y^2 = y
38. Find the locus of a point P(x, y) which is equidistant from the x-axis and the y-axis.
A) y = x
B) y = -x
C) y = x or y = -x
D) x = 0
39. What is the locus of a point that moves such that the difference of its distances from two fixed points is constant?
A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
40. The locus of a point P(x, y) such that the distance from P to the line 3x + 4y - 5 = 0 is 2 is given by:
A) (3x + 4y - 5)^2 = 4(3^2 + 4^2)
B) 3x + 4y - 5 = 2
C) 3x + 4y - 5 = -2
D) 3x + 4y - 5 = 10
41. Find the equation of the locus of a point P(x, y) which is at a distance of 5 units from the point (2, 3).
A) (x - 2)^2 + (y - 3)^2 = 25
B) (x + 2)^2 + (y + 3)^2 = 5
C) (x - 2)^2 + (y - 3)^2 = 5
D) (x + 2)^2 + (y + 3)^2 = 25
42. The locus of a point P(x, y) such that the area of the triangle formed by P and the points (1, 0) and (0, 1) is 1/2 is:
A) x + y = 1
B) x - y = 1
C) x + y = 0
D) x = 1
43. What is the locus of a point that moves such that the sum of its distances from two fixed points is constant?
A) Circle
B) Parabola
C) Hyperbola
D) Ellipse
44. A point P moves such that its distance from point A(1, 0) is always twice its distance from point B(0, 1). Find the locus of P.
A) x^2 + y^2 = 4/3
B) 3x^2 + 3y^2 - 2x - 4y + 1 = 0
C) x^2 - y^2 = 1
D) x + y = 3
45. The equation of the locus of a point (x, y) that is equidistant from the origin (0, 0) and the point (4, 0) is:
A) x = 2
B) y = 2
C) x = 4
D) y = 0
46. What is the locus of a point that moves such that its distance from a fixed point (focus) is always equal to its distance from a fixed line (directrix)?
A) Circle
B) Ellipse
C) Parabola
D) Hyperbola
47. Find the equation of the locus of a point P(x, y) that moves such that PA = PB, where A = (1, 2) and B = (3, 4).
A) x + y = 6
B) x - y = -2
C) 2x + 2y = 12
D) x + y = 5
48. The locus of a point equidistant from two fixed points is:
A) A point
B) A line segment
C) The perpendicular bisector of the line segment joining the two points
D) A circle
49. If a point P(x, y) moves such that its distance from the x-axis is always 5, what is its locus?
A) y = 5
B) x = 5
C) y = -5
D) y = 5 or y = -5
50. What is the locus of a point that moves such that its distance from a fixed line is constant?
A) A circle
B) Two parallel lines
C) A parabola
D) A hyperbola
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