Distance formula and section formula - Question Bank

1. Find the coordinates of the point which divides the line segment joining (1, 5) and (4, 2) externally in the ratio 1:2.
A) (-2, 7)
B) (7, -2)
C) (2, -7)
D) (-7, 2)
2. If a point P divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m:n, then the y-coordinate of P is:
A) (my2 + ny1)/(m+n)
B) (ny2 + my1)/(m+n)
C) (my1 + ny2)/(m+n)
D) (my2 - ny1)/(m-n)
3. What is the distance between the points (x, y) and (-x, -y)?
A) sqrt((2x)^2 + (2y)^2)
B) sqrt(x^2 + y^2)
C) 2*sqrt(x^2 + y^2)
D) 0
4. Find the coordinates of the point which divides the line segment joining (1, 2) and (2, 3) in the ratio 1:2.
A) (5/3, 8/3)
B) (8/3, 5/3)
C) (1, 1)
D) (2, 2)
5. If a point P divides the line segment joining A(x1, y1) and B(x2, y2) such that AP = 3PB, then P divides AB internally in the ratio:
A) 3:1
B) 1:3
C) 3:2
D) 2:3
6. The locus of a point equidistant from two fixed points is:
A) The perpendicular bisector of the line segment joining the two points
B) The line segment joining the two points
C) A circle with the segment as diameter
D) A point
7. Find the coordinates of the point which divides the line segment joining (2, 3) and (5, 6) externally in the ratio 2:1.
A) (8, 9)
B) (9, 8)
C) (-8, -9)
D) (-9, -8)
8. If point P divides AB externally in the ratio m:n (m > n), then the distance AP is what fraction of the total distance AB?
A) m/(m-n)
B) n/(m-n)
C) m/n
D) (m-n)/m
9. Find the coordinates of the point which divides the line segment joining A(5, 9) and B(1, -3) internally in the ratio 1:3.
A) (4, 6)
B) (6, 4)
C) (4, 4)
D) (3, 5)
10. The distance between two points (x1, y1) and (x2, y2) is zero if and only if:
A) x1 = x2 and y1 = y2
B) x1 = -x2 and y1 = -y2
C) x1 = y1 and x2 = y2
D) x1 = x2 or y1 = y2
11. Find the coordinates of the point which divides the line segment joining (1, 2) and (3, 4) in the ratio 1:1.
A) (2, 3)
B) (3, 2)
C) (4, 6)
D) (1, 1)
12. If the origin divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio m:n, then:
A) mx2 + nx1 = 0 and my2 + ny1 = 0
B) nx2 + mx1 = 0 and ny2 + my1 = 0
C) mx1 + nx2 = 0 and my1 + ny2 = 0
D) nx1 + mx2 = 0 and ny1 + my2 = 0
13. What is the distance between the points (a+b, a-b) and (a-b, a+b)?
A) sqrt(8b^2)
B) sqrt(4b^2)
C) 2*sqrt(2)|b|
D) 4|b|
14. Find the ratio in which the point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2).
A) (x-x1)/(x2-x) : (y-y1)/(y2-y)
B) (x-x1)/(x2-x)
C) (y-y1)/(y2-y)
D) (x2-x)/(x-x1) : (y2-y)/(y-y1)
15. If the points (0, 0), (a, b), and (c, d) form a triangle, its area is:
A) 1/2 |ad - bc|
B) |ad - bc|
C) 1/2 (ad - bc)
D) ad - bc
16. The section formula is a generalization of which formula?
A) Midpoint formula
B) Distance formula
C) Centroid formula
D) Area of triangle formula
17. Find the coordinates of the point which divides the line segment joining (1, -3) and (4, 5) externally in the ratio 2:1.
A) (7, 13)
B) (13, 7)
C) (-7, -13)
D) (-13, -7)
18. If a point P divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio k:1, then the coordinates of P are:
A) ((kx2 + x1)/(k+1), (ky2 + y1)/(k+1))
B) ((x1 + kx2)/(k+1), (y1 + ky2)/(k+1))
C) ((kx1 + x2)/(k+1), (ky1 + y2)/(k+1))
D) ((x2 + kx1)/(k+1), (y2 + ky1)/(k+1))
19. What is the distance between the points (-2, 3) and (-2, 8)?
A) 5
B) 11
C) sqrt(25)
D) sqrt(121)
20. Find the coordinates of the point which divides the line segment joining (3, -4) and (-6, 5) internally in the ratio 1:2.
A) (-3, -2)
B) (-2, -3)
C) (3, 2)
D) (2, 3)
21. If point P divides the line segment AB internally in the ratio m:n, then the distance AP is what fraction of the total distance AB?
A) m/(m+n)
B) n/(m+n)
C) m/n
D) (m+n)/m
22. The distance of a point (x, y) from the origin is given by:
A) sqrt(x^2 + y^2)
B) x^2 + y^2
C) |x| + |y|
D) sqrt(|x| + |y|)
23. Find the coordinates of the point which divides the line segment joining A(2, 1) and B(5, 7) in the ratio 1:2.
A) (3, 3)
B) (3, 4)
C) (11/3, 5/3)
D) (11/3, 5)
24. If the points (1, 1), (2, 3), and (3, k) are collinear, find the value of k.
A) 5
B) -5
C) 3
D) 1
25. What is the section formula for internal division?
A) ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
B) ((mx2 - nx1)/(m-n), (my2 - ny1)/(m-n))
C) ((nx2 + mx1)/(n+m), (ny2 + my1)/(n+m))
D) ((mx1 + nx2)/(m+n), (my1 + ny2)/(m+n))
26. Find the value of k if the distance between (k, -1) and (3, 2) is 5.
A) 7 or -1
B) 1 or -7
C) 7
D) -1
27. If a point P divides AB internally in ratio m:n, and externally in ratio p:q, then:
A) m/n = p/q
B) m/n = q/p
C) m*p = n*q
D) m*q = n*p
28. The distance between two points (x1, y1) and (x2, y2) is always non-negative. This is because:
A) We take the square root of a sum of squares
B) We use absolute values
C) The coordinates are always positive
D) The formula involves subtraction
29. Find the coordinates of the point dividing the line segment joining (-4, -2) and (2, 4) externally in the ratio 1:2.
A) (-8, -10)
B) (-10, -8)
C) (8, 10)
D) (10, 8)
30. If the points (a, 0), (0, b), and (x, y) are collinear, then which of the following is true?
A) x/a + y/b = 1
B) x/a + y/b = 0
C) x/b + y/a = 1
D) x/a - y/b = 1
31. A point P divides the line segment joining A(1, 2) and B(6, 7) in the ratio 2:3. Find the coordinates of P.
A) (3, 4)
B) (4, 3)
C) (3, 3)
D) (4, 4)
32. What are the coordinates of the centroid of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3)?
A) ((x1+x2+x3)/3, (y1+y2+y3)/3)
B) ((x1+x2+x3)/2, (y1+y2+y3)/2)
C) ((x1+x2-x3)/3, (y1+y2-y3)/3)
D) ((x1+x2+x3), (y1+y2+y3))
33. Find the distance between the parallel lines 3x + 4y - 5 = 0 and 6x + 8y + 10 = 0.
A) 15/sqrt(5)
B) 15/5
C) sqrt(5)
D) 15
34. If the point (x, y) divides the line joining (x1, y1) and (x2, y2) in the ratio k:1, then x = ?
A) (kx2 + x1)/(k+1)
B) (x1 + kx2)/(k+1)
C) (kx1 + x2)/(k+1)
D) (x2 + kx1)/(k+1)
35. What is the area of the triangle formed by the points (0, 0), (x1, y1), and (x2, y2)?
A) 1/2 |x1*y2 - x2*y1|
B) |x1*y2 - x2*y1|
C) 1/2 (x1*y2 - x2*y1)
D) x1*y2 - x2*y1
36. Find the point which divides the line segment joining (2, -5) and (4, 5) in the ratio 3:2 internally.
A) (14/5, 1)
B) (1, 14/5)
C) (14/5, -1)
D) (1, -14/5)
37. If point C divides the line segment AB internally in the ratio m:n, then the vector equation for C is given by:
A) vec(c) = (n*vec(a) + m*vec(b))/(m+n)
B) vec(c) = (m*vec(a) + n*vec(b))/(m+n)
C) vec(c) = (m*vec(b) - n*vec(a))/(m-n)
D) vec(c) = (n*vec(b) + m*vec(a))/(m+n)
38. What is the perimeter of the triangle with vertices A(0, 0), B(3, 0), and C(0, 4)?
A) 12
B) 10
C) 15
D) 13
39. Find the coordinates of the point which divides the line segment joining A(5, -2) and B(-1, 4) externally in the ratio 3:2.
A) (13, -10)
B) (-10, 13)
C) (13, 10)
D) (-13, 10)
40. If the distance between (a, 2) and (3, 4) is 2, what is the value of 'a'?
A) 3
B) -3
C) 3 or -3
D) 1
41. In what ratio does the point P(2, 3) divide the line segment joining A(-2, 4) and B(3, -7)?
A) 4:5
B) 5:4
C) 3:2
D) 2:3
42. Find the coordinates of the midpoint of the line segment joining A(3, 4) and B(7, 8).
A) (5, 6)
B) (4, 5)
C) (6, 5)
D) (5, 5)
43. What is the distance of the point (6, -2) from the origin (0, 0)?
A) sqrt(40)
B) sqrt(32)
C) sqrt(36)
D) sqrt(64)
44. If a point P divides the line segment joining A(x1, y1) and B(x2, y2) externally in the ratio m:n, what are the coordinates of P?
A) ((mx2 - nx1)/(m-n), (my2 - ny1)/(m-n))
B) ((nx2 - mx1)/(n-m), (ny2 - my1)/(n-m))
C) ((mx1 - nx2)/(m-n), (my1 - ny2)/(m-n))
D) ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
45. The distance formula is derived from which theorem?
A) Pythagorean theorem
B) Thales' theorem
C) Apollonius' theorem
D) Sine rule
46. Find the coordinates of the point which divides the line segment joining A(-1, 7) and B(4, -3) internally in the ratio 2:3.
A) (1, 3)
B) (3, 1)
C) (1, -3)
D) (-1, 3)
47. What is the distance between the points (2, 3) and (5, 7)?
A) 5
B) sqrt(5)
C) 25
D) sqrt(50)
48. If a point P divides the line segment joining A(x1, y1) and B(x2, y2) internally in the ratio m:n, what are the coordinates of P?
A) ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
B) ((nx2 + mx1)/(m+n), (ny2 + my1)/(m+n))
C) ((mx1 + nx2)/(m+n), (my1 + ny2)/(m+n))
D) ((mx2 - nx1)/(m-n), (my2 - ny1)/(m-n))
49. What is the distance between two points (x1, y1) and (x2, y2) in a Cartesian plane?
A) sqrt((x2 - x1)^2 + (y2 - y1)^2)
B) (x2 - x1) + (y2 - y1)
C) |x2 - x1| + |y2 - y1|
D) sqrt((x2 + x1)^2 + (y2 + y1)^2)
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