Question

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

• A. $117$
• B. $153$
• C. $270$
• D. None

Answer 1

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

${}^9{c}_2=\frac{9!}{7!2!}=\frac{9\times 8}{2\times 1}=36$

So, the resulting points of intersection of these pairs $2\times 36=72$ ( 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 ${}^9{c}_2=\frac{9!}{7!2!}=\frac{9\times 8}{2\times 1}=36$

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 $9\times 9=81$

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 $81\times 2=162$

So after solving these cases, we get the total number of points as,

$\therefore 72+36+162=270$

Correct answer is C.