Question

The maximum number of points of intersection of 4 circle and 4 straight lines is
A.25
B.50
C.56
D.72

Answer 1

Hint- For at least one intersection we need two lines or two circles or one circle and one straight line. First find out the maximum intersection between lines and circles separately. Then generalize it to get the required result.

Complete step-by-step answer:
In the given 4 lines we need to select at least two lines for one intersection point.
Maximum number of points in 4 straight lines is${}^{4}{{C}_{2}}\times 1=\dfrac{4!}{2!.2!}=6$.
There will be a maximum 6 intersection points between 4 lines.
Between 2 circles there are 2 intersection points.
Maximum number of points in 4 circles is ${}^{4}{{C}_{2}}\times 2=\dfrac{4!}{2!.2!}\times 2=12$
Between one line and one circle, there are maximum 2 intersection points.
Maximum number of intersection points between 1(out of 4) circle and 1(out of 4) line is,
${}^{4}{{C}_{1}}\times {}^{4}{{C}_{1}}\times 2=32$
Thus, maximum number of intersection points between 4 circle and 4 lines is,
6 + 12 +32 =50.
Option (B) is correct.

Note- We can also draw lines and circle separately and count the intersection point. This given statement is similar to selecting r number of objects from n objects ${}^{n}{{C}_{r}}$.