JEE Main - Distance of a point from a line and coordinates of triangle centres Online Test (MCQ Quiz)

1. What is the formula for the distance of a point (x₁, y₁) from the line Ax + By + C = 0?

A) |Ax₁ + By₁ + C| / √(A² + B²) B) |Ax₁ + By₁ + C| / (A² + B²) C) (Ax₁ + By₁ + C) / √(A² + B²) D) |Ax₁ - By₁ + C| / √(A² + B²)

2. Find the distance of the point (3, 4) from the line 3x + 4y - 10 = 0.

A) 1 B) 2 C) 3 D) 4

3. If the distance of the point (k, 2) from the line 4x - 3y + 10 = 0 is 2 units, find the possible values of k.

A) k = 2 or k = -8 B) k = -2 or k = 8 C) k = 2 or k = 8 D) k = -2 or k = -8

4. 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) ((x₁x₂x₃)/3, (y₁y₂y₃)/3)

5. Find the centroid of the triangle with vertices A(1, 2), B(3, 4), and C(5, 6).

A) (3, 4) B) (4, 5) C) (5, 4) D) (3, 3)

6. The centroid of a triangle divides each median in the ratio:

A) 2:1 B) 1:2 C) 1:1 D) 3:1

7. What are the coordinates of the orthocenter of a triangle?

A) Intersection of altitudes B) Intersection of medians C) Intersection of perpendicular bisectors D) Intersection of angle bisectors

8. For a triangle with vertices A(0, 0), B(3, 0), and C(0, 4), find the coordinates of the orthocenter.

A) (0, 0) B) (1, 1) C) (2, 2) D) (3, 4)

9. What are the coordinates of the circumcenter of a triangle?

A) Intersection of perpendicular bisectors B) Intersection of altitudes C) Intersection of medians D) Intersection of angle bisectors

10. Find the circumcenter of the triangle with vertices A(0, 0), B(2, 0), and C(0, 2).

A) (1, 1) B) (0, 0) C) (2, 2) D) (1, 0)

11. What are the coordinates of the incenter of a triangle?

A) Intersection of angle bisectors B) Intersection of altitudes C) Intersection of medians D) Intersection of perpendicular bisectors

12. The incenter of a triangle is equidistant from:

A) The sides of the triangle B) The vertices of the triangle C) The medians of the triangle D) The altitudes of the triangle

13. For a triangle with vertices A(0, 0), B(3, 0), and C(0, 4), find the coordinates of the incenter.

A) (2, 2) B) (1, 1) C) (3, 4) D) (0, 0)

14. The distance of a point from a line is zero if and only if:

A) The point lies on the line B) The point is the origin C) The line passes through the origin D) The line is parallel to the x-axis

15. Consider the line 5x - 12y + 26 = 0. Find the distance of the origin from this line.

A) 26/13 B) 13/26 C) 26/5 D) 12/26

16. If two lines are parallel, Ax + By + C₁ = 0 and Ax + By + C₂ = 0, what is the distance between them?

A) |C₁ - C₂| / √(A² + B²) B) |C₁ + C₂| / √(A² + B²) C) |C₁ - C₂| / (A² + B²) D) |C₁ - C₂| / |A + B|

17. Find the distance between the parallel lines 3x + 4y - 5 = 0 and 3x + 4y + 10 = 0.

A) 15/5 B) 5/15 C) 15/25 D) 25/15

18. The orthocenter, centroid, and circumcenter of a triangle are collinear. This line is known as the:

A) Euler line B) Median line C) Altitude line D) Bisector line

19. In an equilateral triangle, which of the following points coincide?

A) Centroid, Orthocenter, Circumcenter, Incenter B) Centroid and Orthocenter only C) Circumcenter and Incenter only D) Orthocenter and Centroid only

20. For a right-angled triangle, the circumcenter is the:

A) Midpoint of the hypotenuse B) Vertex with the right angle C) Intersection of the altitudes D) Centroid

21. Find the orthocenter of a triangle with vertices (0, 0), (a, 0), and (0, b).

A) (0, 0) B) (a, b) C) (a/2, b/2) D) (a, 0)

22. What is the distance of the point P(2, 3) from the line y = 5?

A) 2 B) 3 C) 5 D) 0

23. What is the distance of the point P(2, 3) from the line x = -1?

A) 3 B) 2 C) 1 D) 0

24. The coordinates of the vertices of a triangle are (1, 2), (3, 5), and (4, 1). Find the coordinates of its centroid.

A) (8/3, 8/3) B) (2, 3) C) (3, 4) D) (4, 2)

25. If the centroid of a triangle with vertices (a, 1), (2, b), and (3, 4) is (2, 3), find a and b.

A) a=1, b=4 B) a=2, b=3 C) a=3, b=2 D) a=4, b=1

26. The circumcenter of a triangle is the intersection of:

A) Perpendicular bisectors of the sides B) Medians C) Altitudes D) Angle bisectors

27. Find the circumcenter of the triangle with vertices A(1, 0), B(-1, 0), and C(0, √3).

A) (0, 0) B) (1, 1) C) (0, 1) D) (1, 0)

28. The incenter of a triangle is the intersection of:

A) Angle bisectors B) Perpendicular bisectors C) Medians D) Altitudes

29. The distance of the point (x₁, y₁) from the line Ax + By + C = 0 is positive if:

A) The point and the origin lie on opposite sides of the line B) The point and the origin lie on the same side of the line C) The line passes through the origin D) The line is horizontal

30. Find the distance of the point (6, 8) from the line 4x + 3y - 12 = 0.

A) 60/5 B) 5/60 C) 12/5 D) 60/13

31. What is the distance of the point (1, -2) from the line 3x - 4y + 5 = 0?

A) 18/5 B) 5/18 C) 18/13 D) 13/18

32. The orthocenter of the triangle with vertices (0, 0), (1, 0), and (0, 1) is:

A) (0, 0) B) (1, 1) C) (1/2, 1/2) D) (0, 1)

33. For a triangle ABC, if A=(0,0), B=(3,0), C=(0,4), find the coordinates of its circumcenter.

A) (3/2, 2) B) (1, 1) C) (0, 0) D) (2, 3)

34. Find the incenter of an equilateral triangle with vertices at (0, 0), (a, 0), and (a/2, a√3/2).

A) (a/2, a√3/6) B) (a/2, a√3/2) C) (0, 0) D) (a, 0)

35. The distance between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is given by:

A) |C₁ - C₂| / √(A² + B²) B) |C₁ + C₂| / √(A² + B²) C) |C₁ - C₂| / (A² + B²) D) √(C₁² - C₂²) / √(A² + B²)

36. Find the distance between the parallel lines y = 2x + 4 and y = 2x + 9.

A) 5/√5 B) √5/5 C) 5/2 D) 2/5

37. The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is G. If G = (0,0), then:

A) x₁ + x₂ + x₃ = 0 and y₁ + y₂ + y₃ = 0 B) x₁ + x₂ + x₃ = 3 and y₁ + y₂ + y₃ = 3 C) x₁ + x₂ + x₃ = 1 and y₁ + y₂ + y₃ = 1 D) x₁x₂x₃ = 0 and y₁y₂y₃ = 0

38. If the orthocenter of a triangle with vertices (0, 0), (a, 0), and (0, b) is (0, 0), then:

A) The triangle is right-angled at the origin B) The triangle is equilateral C) The triangle is isosceles D) The triangle is scalene

39. The circumcenter of a triangle with vertices (0, 0), (2a, 0), and (a, a√3) is:

A) (a, a/√3) B) (a, 0) C) (0, 0) D) (a, a)

40. Find the distance of the point (2, -1) from the line 2x + 5y + 6 = 0.

A) 0/√29 B) √29/0 C) 10/√29 D) 2/√29

41. The distance of the point (a cosθ, a sinθ) from the line x cosθ + y sinθ - a = 0 is:

A) 0 B) a C) 2a D) a/2

42. If the distance of point P from the line √3x + 4y + 5 = 0 is 2, then P can be:

A) A point such that |3√3 + 8 + 5| = 10 B) A point such that |3√3 + 8 + 5| = 20 C) A point such that |3√3 + 8 + 5| = 5 D) A point such that |3√3 + 8 + 5| = 2

43. The line joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. This is a property related to:

A) Midpoint Theorem B) Centroid Theorem C) Pythagorean Theorem D) Sine Rule

44. The coordinates of the vertices of a triangle are (3, 4), (5, -2), and (1, -6). Find the coordinates of the centroid.

A) (3, -4/3) B) (9, -4) C) (3, 0) D) (4, 1)

45. Find the orthocenter of a triangle with vertices A(0, 0), B(a, 0), and C(a/2, a√3/2).

A) (a/2, a√3/6) B) (0, 0) C) (a, 0) D) (a/2, 0)

46. The circumcenter of a triangle with vertices (0, 0), (b, 0), and (0, c) is:

A) (b/2, c/2) B) (0, 0) C) (b, c) D) (b/2, 0)

47. Find the incenter of the triangle with vertices (0, 0), (3, 0), and (0, 4).

A) (1, 1) B) (2, 2) C) (3, 4) D) (0, 0)

48. The distance of the point (2, 3) from the line x/3 + y/4 = 1 is:

A) 12/5 B) 5/12 C) 1 D) 0

49. If the distance of the point (1, k) from the line 3x + 4y - 12 = 0 is 3, find the value of k.

A) k = 0 or k = -12/2 B) k = 3 or k = -9 C) k = 0 or k = -6 D) k = 1 or k = -5

50. The distance of the point (x, y) from the line Ax + By + C = 0 is proportional to:

A) Ax + By + C B) (Ax + By + C)² C) √(Ax + By + C) D) 1 / (Ax + By + C)

Distance of a point from a line and coordinates of triangle centres - Online Test

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