JEE Main - Distance formula and section formula Online Test (MCQ Quiz)

1. 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)

2. 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))

3. What is the distance between the points (2, 3) and (5, 7)?

A) 5 B) sqrt(5) C) 25 D) sqrt(50)

4. 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)

5. The distance formula is derived from which theorem?

A) Pythagorean theorem B) Thales' theorem C) Apollonius' theorem D) Sine rule

6. 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))

7. 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)

8. 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)

9. 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

10. 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

11. 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)

12. 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

13. 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)

14. 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)

15. 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

16. 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)

17. 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

18. 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))

19. 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)

20. 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

21. 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)

22. 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

23. 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

24. 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

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. 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

27. 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)

28. 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|)

29. 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

30. 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)

31. What is the distance between the points (-2, 3) and (-2, 8)?

A) 5 B) 11 C) sqrt(25) D) sqrt(121)

32. 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))

33. 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)

34. The section formula is a generalization of which formula?

A) Midpoint formula B) Distance formula C) Centroid formula D) Area of triangle formula

35. 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

36. 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)

37. 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|

38. 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

39. 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)

40. 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

41. 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)

42. 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

43. 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)

44. 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

45. 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

46. 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)

47. 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

48. 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)

49. 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)

Distance formula and section formula - Online Test

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