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Question

The greatest possible number of points of intersection of 9 different straight lines & 9 different circles in a plane is?

A
117
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B
153
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C
270
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D
None
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Solution

The correct option is A 117
Case .1 : Intersection of all circle
There are 9 circle any two pairs of circle can intersect most of 2 points. So the maximum number of points of intersection is when all the possible combinations of pairs of circles intersect at two points.
i.e. , the number of such combination of circles are
9c2=9!7!2!=9×82×1=36
So, the resulting points of intersection of these pairs 2×36=72 ( we are multiplying by 2 because each pair can intersect at 2 points)
Case 2: Intersection of all circle
There are nine lines and each pair can intersect utmost at only one points. So again the number of such combination are 9c2=9!7!2!=9×82×1=36
So, the new set of point is 36
Case 3: Intersection of a line and a circle
A line can intersect a circle again in a maximum of two points. Assuming all lines intersect all circles at 2 points each, the number of such combination of a line and a circle are 9×9=81
Explanation: For each line, there are. 9 circles i.e. choose from to intersect, and there are a total of 9 lines. So or all the lines and circles together, there are a total of. 81 combinations possible . Now, for these 81 combinations, the total number of unique points created are 81×2=162
So after solving these cases, we get the total number of points as,
72+36+162=270
Correct answer is C.

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