Locus and its equation - One Line Questions

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