Question

Total number of regions in which 'n' coplanar lines can divide the plane, it is known that no two lines are parallel and no three of them are concurrent, is equal to

• A. \(\frac{1}{2}(n^2 + n + 2)\)
• B. \(\frac{1}{2}(n + 3n^2)\)
• C. \(\frac{1}{2}(3n + n^2)\)
• D. \((n^2 - n + 2)\)

Hint:

The maximum number of regions from n lines is \(\frac{n(n+1)}{2} + 1\).

Answer 1

The Correct Option is A

To determine the total number of regions into which 'n' coplanar lines can divide the plane, given that no two lines are parallel and no three lines are concurrent, we use a specific mathematical formula. The correct formula for calculating this is:

\(\frac{1}{2}(n^2 + n + 2)\)

Let's break down the reasoning:

1. When \(n = 0\) (no lines), the plane is undivided, resulting in 1 region.
2. When \(n = 1\) (one line), it divides the plane into 2 regions.
3. When \(n = 2\) (two lines are intersecting), it divides the plane into 4 regions.
4. When \(n = 3\), the lines can intersect such that the regions increase to 7.
5. The pattern continues as more lines are added. Therefore, we need to derive a general formula that encapsulates this growth pattern.

To understand how this pattern grows, consider:

• Each new line can potentially intersect all previous lines, thus increasing the number of regions.
• The formula that correctly models this growth is derived from combinatorial mathematics and is given as: \(R = \frac{1}{2}(n^2 + n + 2)\).

Let’s verify this with a few examples:

• For \(n = 0\): \(\frac{1}{2}(0^2 + 0 + 2) = 1\). Correct.
• For \(n = 1\): \(\frac{1}{2}(1^2 + 1 + 2) = 2\). Correct.
• For \(n = 2\): \(\frac{1}{2}(2^2 + 2 + 2) = 4\). Correct.
• For \(n = 3\): \(\frac{1}{2}(3^2 + 3 + 2) = 7\). Correct.

The formula holds for all values of \(n\) that meet the given conditions (no two lines parallel, no three concurrent).

Therefore, the correct answer is \(\frac{1}{2}(n^2 + n + 2)\).