Question:
In a plane there are 37 straight lines of which 13 pass through point A and 11 pass through point B. Moreover, no three lines (apart from the lines passing through A and B) pass through same point and no two lines are parallel. What is the number of points of intersection of the straight lines?
In a plane there are 37 straight lines of which 13 pass through point A and 11 pass through point B. Moreover, no three lines (apart from the lines passing through A and B) pass through same point and no two lines are parallel. What is the number of points of intersection of the straight lines?
Show Hint
Start with the total number of intersections for \(n\) lines if no two are parallel and no three are concurrent: \(\binom{n}{2}\).
If \(k\) lines are concurrent at a point, they create 1 intersection point instead of \(\binom{k}{2}\). The reduction is \(\binom{k}{2} - 1\).
Subtract the "lost" intersections for each point of concurrency and add back the single point of concurrency itself.
Total = \(\binom{N_{total}}{2} - \sum (\binom{k_i}{2} - 1)\) where \(k_i\) is number of lines concurrent at point i.
Updated On: Mar 18, 2026
- \( {}^{37}C_2 \)
- \( {}^{37}C_2 - {}^{13}C_2 - {}^{11}C_2 \)
- \( {}^{37}C_2 - {}^{13}C_2 - {}^{11}C_2 + 2 \)
- \( {}^{37}C_2 - 2 \)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
We are given:
- Total of 37 lines.
- 13 lines pass through point A.
- 11 lines pass through point B.
- Points A and B are distinct.
- No three lines meet at any point other than A or B.
- No two lines are parallel.
We aim to compute the total number of distinct intersection points.
🔹 Total intersections without constraints
If all 37 lines intersect pairwise without concurrency:
\[ \text{Total Intersections} = \binom{37}{2} = \frac{37 \cdot 36}{2} = 666 \]
🔹 Adjust for concurrency at point A
\[ \binom{13}{2} = \frac{13 \cdot 12}{2} = 78 \] But since all 13 lines intersect at point A, they produce only 1 intersection.
So, we subtract:
\[ 78 - 1 = 77 \]
🔹 Adjust for concurrency at point B
\[ \binom{11}{2} = \frac{11 \cdot 10}{2} = 55 \] But all 11 lines intersect at point B, producing only 1 intersection.
So, we subtract:
\[ 55 - 1 = 54 \]
✅ Final Count
\[ \text{Valid Intersections} = 666 - 77 - 54 = \boxed{535} \]
🧠 Assumptions Validated
- No triple intersections outside A and B ✔️
- Lines through A only intersect at A, and similarly for B ✔️
- No overlaps beyond A and B ✔️
- Possibility of one line passing through both A and B is handled correctly ✔️
🎯 Final Answer:
\[ \boxed{535 \text{ distinct points of intersection}} \]
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