Question

There are 9 straight lines of which 5 are concurrent at a point and other 4 are concurrent and no two of these 9 lines are parallel then number points of intersection is?
A. 20
B. 22
C. 36
D. 38

Answer 1

Hint: In the above given question, we are given a number of 9 straight lines out of which given that 5 are concurrent at one point while the other 4 are also concurrent at another point. It is also given in the question that no two lines out of 9 given straight lines are parallel. That means all the 9 lines are non-parallel and hence, each straight line intersects the other remaining straight lines at a point. We have to determine the total number of points of intersections of these 9 straight lines.

Complete step by step answer:
We are given 9 straight lines out of which 5 are concurrent at one point and the other 4 straight lines are concurrent at other point. Also, no two straight lines are parallel i.e. they all intersect each other. We have to find the total number of points of intersection. Now, since we have to find the intersection of each pair of straight lines i.e. every 2 straight lines out of the 9 given straight lines.Therefore, the total number of points of intersection of 9 non-parallel lines is given by the expression \[^9{C_2}\]. But it is given that 5 of these lines are concurrent, therefore there will be \[^5{C_2}\] less points of intersections.Similarly, there are 4 other straight lines which are concurrent at other points, so there will be a loss of \[^4{C_2}\] more points of intersections. But these two concurrent points will also add up in the sum of points of intersection.

Now, the total number of points of intersection will be given by the resulting sum of all these number of points of intersection, that is
\[{ \Rightarrow ^9}{C_2}{ - ^5}{C_2}{ - ^4}{C_2} + 2\]
That gives us,
\[ \Rightarrow \left( {\dfrac{9}{2} \cdot \dfrac{8}{1}} \right) - \left( {\dfrac{5}{2} \cdot \dfrac{4}{1}} \right) - \left( {\dfrac{4}{2} \cdot \dfrac{3}{1}} \right) + 2\]
Evaluating the above sum, we get
\[ \Rightarrow 36 - 10 - 6 + 2\]
That is,
\[ \Rightarrow 22\]
That is the required number of points of intersection. Therefore, the number of points of intersection is \[22\].

Hence, the correct option is B.

Note: When two or more than two straight lines intersect each other at a single point, then the set of these straight lines is known as the set of concurrent lines. Whereas, in case of intersecting lines, it concerns only two straight lines. The common examples of concurrent lines are, the three medians, angle bisectors, perpendicular bisectors, or the altitudes of a triangle.