The maximum possible number of points of intersection of 8 straight lines and 4 circles is

Options

A. 164
B. 76
C. 104
D. 32

Answer

To find the maximum possible number of points of intersection of 8 straight lines and 4 circles, we will break down the problem into three parts: intersections among lines, intersections among circles, and intersections between lines and circles.

### Step 1: Calculate the intersection points among the straight lines.
The maximum number of intersection points formed by \( n \) straight lines is given by the formula:
\[ \text{Number of intersections} = \binom{n}{2} = \frac{n(n-1)}{2} \]
For 8 lines:
\[ \text{Number of intersections} = \binom{8}{2} = \frac{8 \times 7}{2} = 28 \]

### Step 2: Calculate the intersection points between lines and circles.
Each line can intersect a circle at most at 2 points. Therefore, for 8 lines and 4 circles, the maximum number of intersection points is:
\[ \text{Number of intersections} = \text{Number of lines} \times \text{Number of circles} \times 2 = 8 \times 4 \times 2 = 64 \]

### Step 3: Calculate the intersection points among the circles.
The maximum number of intersection points formed by \( m \) circles is given by the formula:
\[ \text{Number of intersections} = \binom{m}{2} \times 2 \]
For 4 circles:
\[ \text{Number of intersections} = \binom{4}{2} \times 2 = \frac{4 \times 3}{2} \times 2 = 6 \times 2 = 12 \]

### Step 4: Sum all the intersection points.
Now, we add the intersection points from all three cases:
\[ \text{Total intersections} = \text{Intersections among lines} + \text{Intersections between lines and circles} + \text{Intersections among circles} \]
\[ \text{Total intersections} = 28 + 64 + 12 = 104 \]

### Final Answer:
The maximum possible number of points of intersection of 8 straight lines and 4 circles is **104**.

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