Geometry Concepts and Questions : Number of Lines from n Points on a Plane – Must Practice Set for CAT, XAT & Other MBA Exams
May 29, 2024 2025-05-30 13:45Geometry Concepts and Questions : Number of Lines from n Points on a Plane – Must Practice Set for CAT, XAT & Other MBA Exams
Geometry Concepts and Questions : Number of Lines from n Points on a Plane – Must Practice Set for CAT, XAT & Other MBA Exams
Amiya Kumar Number of Lines from n Points on a Plane – Geometry Concept
In geometry, one of the most common problems is to find how many distinct straight lines can be formed using a given number of points on a plane. This concept is useful for competitive exams like CAT, XAT, SSC, and NTSE.
🔹 Basic Concept: No Three Points are Collinear
If no three points are collinear, then any two points will uniquely form a line. So, the number of lines formed from n such points is:
This is the maximum number of lines that can be formed from n points.
Example 1:
If there are 5 points and no three are collinear, then:
🔹 Concept: Some Points are Collinear
If some points are collinear, they lie on the same line. So instead of forming multiple lines, they contribute only one line.
Let's say m points are collinear. Then the total number of lines is:
Example 2:
Out of 7 points, 4 are collinear. Find total number of lines.
🔹 Generalized Concept for Multiple Collinear Groups
If there are multiple sets of collinear points, say m, p, q... collinear points in separate groups, the formula becomes:
Example 3:
In a plane, 10 points are given, out of which 4 are collinear and 3 are collinear in another set. Find total number of lines.
📌 Final Notes
- If all n points are collinear, only 1 line is formed.
- If no three points are collinear, maximum lines = \( \frac{n(n-1)}{2} \)
- Always subtract lines from collinear subsets and add 1 line per collinear group.
📐 Number of Lines from n Points – Practice Questions for CAT/XAT
Click below to reveal answers and explanations. Concept based on combinations and collinear point deductions.
1) From 6 points on a plane, where 3 points are collinear, how many distinct lines can be formed?
- (a) 12
- (b) 13
- (c) 14
- (d) 15
Answer: (b) 13
Solution: \( \binom{6}{2} - \binom{3}{2} + 1 = 15 - 3 + 1 = 13 \)
2) If no three of the 10 points are collinear, how many distinct lines can be formed?
- (a) 40
- (b) 44
- (c) 45
- (d) 48
Answer: (c) 45
Solution: \( \binom{10}{2} = \frac{10 \cdot 9}{2} = 45 \)
3) From 6 points on a plane, where 4 and 3 points are collinear in two separate groups, how many lines can be formed?
- (a) 6
- (b) 7
- (c) 8
- (d) 9
Answer: (c) 8
Solution: \( \binom{6}{2} - \binom{4}{2} - \binom{3}{2} + 2 = 15 - 6 - 3 + 2 = 8 \)
4) If no three of the 14 points are collinear, how many lines can be formed?
- (a) 90
- (b) 91
- (c) 92
- (d) 105
Answer: (b) 91
Solution: \( \binom{14}{2} = \frac{14 \cdot 13}{2} = 91 \)
5) From 13 points on a plane, where 4 and 3 are collinear in two different lines, how many distinct lines can be drawn?
- (a) 69
- (b) 70
- (c) 71
- (d) 72
Answer: (c) 71
Solution: \( \binom{13}{2} - \binom{4}{2} - \binom{3}{2} + 2 = 78 - 6 - 3 + 2 = 71 \)
6) If no three of the 7 points are collinear, how many lines can be formed?
- (a) 21
- (b) 20
- (c) 22
- (d) 19
Answer: (a) 21
Solution: \( \binom{7}{2} = \frac{7 \cdot 6}{2} = 21 \)
7) From 8 points on a plane, where 3 are collinear, how many lines can be formed?
- (a) 27
- (b) 26
- (c) 28
- (d) 25
Answer: (a) 27
Solution: \( \binom{8}{2} - \binom{3}{2} + 1 = 28 - 3 + 1 = 26 \)
8) From 11 points, where 5 are collinear, how many distinct lines can be formed?
- (a) 50
- (b) 51
- (c) 52
- (d) 53
Answer: (b) 51
Solution: \( \binom{11}{2} - \binom{5}{2} + 1 = 55 - 10 + 1 = 46 \)
9) From 12 points, where 3 and 4 are collinear in two groups, how many lines are possible?
- (a) 64
- (b) 66
- (c) 67
- (d) 68
Answer: (c) 67
Solution: \( \binom{12}{2} - \binom{3}{2} - \binom{4}{2} + 2 = 66 - 3 - 6 + 2 = 59 \)
10) If all 9 points are collinear, how many lines can be formed?
- (a) 36
- (b) 9
- (c) 1
- (d) 0
Answer: (c) 1
Solution: All points are collinear ⇒ only 1 line.
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