Question

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

• A. 117
• B. 153
• C. 270
• D. none of these

Answer 1

The correct option is C 270

Case $1:$Intersection of all circle

There are nine circle.Any two pairs of circles can intersect atmost at $2$ 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 combinations 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 are $2\times 36=72$(We are multiplying by $2$ because each pair can intersect at two points.)

Case$2:$Intersection of all circle:

There are nine lines and each pair can intersect atmost at only one point.So again the number of such combinations are ${}^9{C}_2=\frac{9!}{7!2!}=\frac{9\times 8}{2\times 1}=36$.So the new set of points 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 combinations of a line and a circle are $9\times 9=81$

Explanation:For each line, there are $9$ circles to choose from to intersect.And there are a total of $9$ lines.So for 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,

$72+36+162=270$