Problem 72 - Algebra for JEE ADVANCED

Question

\(n\) lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent. The number of different points at which these lines will cut is a. \(\sum_{k=1}^{n-1} k\) b. \(n(n-1)\) c. \(n^{2}\) d. none of these

Step-by-Step Solution

Answer

None of the options exactly match \( \frac{n(n-1)}{2} \), which is the correct solution.

Step 1: Understanding the Problem

We have a problem where we want to determine the number of intersection points formed when n lines are drawn such that no two lines are parallel and no three are concurrent. We need to calculate the answer and match it with one of the given options.

Step 2: Identify the Formula

For n lines drawn in such a manner, each pair of lines intersects at a unique point. Thus, the formula to calculate the number of intersection points is given by \( \binom{n}{2} \), which represents the number of ways to choose 2 lines out of n lines to form an intersection.

Step 3: Calculate \( \binom{n}{2} \)

Using the combination formula, we know \( \binom{n}{2} = \frac{n(n-1)}{2} \). This is derived by calculating the possible pairs of lines from n lines. Each pair contributes one intersection point.

Step 4: Match with Given Options

The calculated number of intersection points is \( \frac{n(n-1)}{2} \). Looking at the given options, none of the options directly express \( \frac{n(n-1)}{2} \), rather option b is given as \( n(n-1) \). In this problem context, especially when checking for common problem-solving methods, \( \frac{n(n-1)}{2} \) is the correct count but it doesn't strictly match \( n(n-1) \) without considering additional context or assumptions.

Key Concepts

Intersection of Lines

When we talk about the intersection of lines in a plane, we mean the specific points where these lines cross each other. Imagine having several lines laid out on a flat surface, such as your desk. Where each pair of lines cross each other, an intersection point is formed.

It's crucial to understand the condition given in problems like these: no two lines are parallel, and no three lines meet at a single point (concurrent). Why is this important? If two lines were parallel, they'd never intersect, reducing the total number of intersection points. Similarly, if three lines were concurrent, they'd only create one intersection point instead of three. Ensuring no parallels and no concurrency maximizes the number of intersection points.

The beauty here is that each unique pair of lines creates one intersection point. Therefore, the problem effectively becomes one of counting how many unique pairs can be formed from the given lines. This leads us to the concept of binomial coefficients, which helps us calculate these combinations.

Binomial Coefficient

A binomial coefficient, often read as "n choose k," is a fundamental concept in combinatorics. It represents the number of ways to choose k items from a total of n without regard to the order of selection. In symbols, this is denoted as \( \binom{n}{k} \).

In our problem, we use \( \binom{n}{2} \), since we are interested in forming pairs (k=2) of lines from a total (n) number of lines. This is computed using the formula:

• \( \binom{n}{2} = \frac{n(n-1)}{2} \)

Here, \( n(n-1) \) computes the total number of unordered pairs, while dividing by 2 accounts for the order not mattering in pair selection.

The binomial coefficient tells us how many unique intersections there will be for our set of lines. This combinatorial approach ensures all line pairs are counted precisely once, reflecting each intersection formed perfectly.

Plane Geometry

Plane geometry involves the study of figures like lines, circles, and polygons that lie in a flat plane. This branch of geometry handles how these figures interact, such as intersecting lines or combining shapes to understand new forms.

When handling problems involving a plane, like determining intersection points of lines, it's crucial to leverage geometric principles and combinatorial tactics for accurate solutions. Here, lines in a plane are assumed to extend infinitely in both directions and interact at points when two lines cross.

Imagine placing several pencils on a flat table, each crossing the others at different points. This physical setup mimics lines in plane geometry, demonstrating concepts such as intersection and how placing these lines differently (ensuring no parallelism or concurrency) affects their intersection count.

Utilizing plane geometry concepts enables us to explore configurations beyond typical figures, revealing new insights into how these straight trajectories interact and the mathematical elegance they exhibit in a flat, two-dimensional space.