On a plane, Bob draws 33 lines, out of which 11 pass through point P and 9 passes through point Q. There are no three lines that pass through one point on this plane. Also, there is no line that passes through both points P and Q and no two lines are parallel. He wants to calculate the total number of points of intersection. How many such points exist on this plane?
Question
On a plane, Bob draws 33 lines, out of which 11 pass through point P and 9 passes through point Q. There are no three lines that pass through one point on this plane. Also, there is no line that passes through both points P and Q and no two lines are parallel. He wants to calculate the total number of points of intersection. How many such points exist on this plane?
Solution
To calculate the total number of points of intersection, we need to understand that any two lines intersect at one point unless they are parallel. Since no two lines are parallel in this case, we can use the formula n(n-1)/2 to calculate the number of intersections, where n is the number of lines.
First, calculate the number of intersections for the 33 lines: 33*(33-1)/2 = 528.
Next, subtract the intersections at points P and Q. Since no line passes through both points P and Q, the intersections at P and Q are counted twice in the initial calculation.
The number of intersections at point P (with 11 lines passing through) is: 11*(11-1)/2 = 55. The number of intersections at point Q (with 9 lines passing through) is: 9*(9-1)/2 = 36.
Subtract these from the total number of intersections: 528 - 55 - 36 = 437.
So, there are 437 points of intersection on the plane.
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