Problem 64 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Consider the set of all lines \(p x+q y+r=0\) such that \(3 \mathrm{p}+2 \mathrm{q}+4 \mathrm{r}=0 .\) Which one of the following statements is true? \(\quad\) [Jan. 9,2019 (I)] (a) The lines are concurrent at the point \(\left(\frac{3}{4}, \frac{1}{2}\right)\). (b) Each line passes through the origin. (c) The lines are all parallel. (d) The lines are not concurrent.

Step-by-Step Solution

Answer

(a) The lines are concurrent at the point \(\left(\frac{3}{4}, \frac{1}{2}\right)\).

Step 1: Analyze the Condition for the Lines

We are given the set of lines described by the equation \( p x + q y + r = 0 \) with the condition \( 3p + 2q + 4r = 0 \). This is an extra constraint on the parameters of the lines.

Step 2: Assume Lines Pass Through a Common Point

For the lines to be concurrent at a point \((x_0, y_0)\), they must satisfy both equations simultaneously: \( p x_0 + q y_0 + r = 0 \) and \( 3p + 2q + 4r = 0 \). We test this by substituting \((x_0, y_0) = \left(\frac{3}{4}, \frac{1}{2}\right)\) into the first equation.

Step 3: Substitute the Point into the Line Equation

Substituting \(x_0 = \frac{3}{4}\) and \(y_0 = \frac{1}{2}\) into the line equation, we get: \( p\frac{3}{4} + q\frac{1}{2} + r = 0 \). Multiply through by 4 to clear fractions, yielding: \( 3p + 2q + 4r = 0 \).

Step 4: Check Compatibility of the Conditions

Notice that the equation \( 3p + 2q + 4r = 0 \) obtained from the substitution is indeed the given condition. Thus, every line in this set passes through the point \( \left( \frac{3}{4}, \frac{1}{2} \right) \), making the lines concurrent at this point.

Step 5: Evaluate the Given Options

Given the analysis:(a) is true since they all intersect at \( \left( \frac{3}{4}, \frac{1}{2} \right) \). (b) is false because not all lines necessarily pass through the origin. (c) is false because the lines are not parallel—they intersect at a specific point. (d) is false because the lines are concurrent. Hence, option (a) is the correct statement.

Key Concepts

Concurrent Lines

In geometry, concurrent lines are lines that intersect or meet at a single point, known as the point of concurrency. This concept is crucial in solving many geometrical problems because it simplifies the conditions you need to verify.
Understanding concurrent lines often involves finding a point where several lines pass through simultaneously.

• Consider three or more lines in a plane. If there is a common point that all lines pass through, the lines are concurrent.
• Finding whether lines are concurrent often involves solving a system of equations to determine if there is a solution common to all lines involved.

In our problem, to determine concurrency, we checked if all lines described by the equation \[ p x + q y + r = 0 \]correctly intersect at the point \( \left( \frac{3}{4}, \frac{1}{2} \right) \). By substituting this point into the line equation and checking with the conditions given, we found that the lines do indeed intersect at this single point, confirming they are concurrent.

Systems of Equations

A system of equations is a collection of two or more equations with a set of variables. The goal is to find the values for the variables that satisfy all equations in the system simultaneously.
Solving systems of equations involves finding a common solution for these equations.

• There are different methods to solve systems, such as substitution, elimination, and using matrices.
• The solution can be a point where all the equations in the system meet, often relevant in geometry.

In terms of our line set, we used the system of equations formed by \[ 3p + 2q + 4r = 0 \] and \[ p \frac{3}{4} + q \frac{1}{2} + r = 0 \] to show that lines pass through a particular point, making them concurrent. By resolving this system, we identify the necessary condition for concurrency.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, involves using algebraic equations to describe geometrical principles and figures. It provides a bridge between algebra and geometry, making it easier to analyze shapes and their positions using coordinates.
For lines, coordinate geometry is primarily about understanding their equations and how they interact in a coordinate plane.

• A line in a coordinate plane can be represented by the equation \( p x + q y + r = 0 \).
• Understanding the geometric implications of such equations, like slopes and intercepts, is key to solving problems involving lines.

In our specific problem, coordinate geometry allowed us to express the lines and examine their concurrency through algebraic conditions. The method illustrated how geometric ideas can be effectively handled using equations, leading to analytical solutions such as determining where lines intersect true to their algebraic formulations.