Problem 73 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation of the straight line passing through the point \((4,3)\) and making intercepts on the co-ordinate axes whose sum is \(-1\) is (A) \(\frac{x}{2}+\frac{y}{3}=-1\) and \(\frac{x}{-2}+\frac{y}{1}=-1\) (B) \(\frac{x}{2}-\frac{y}{3}=-1\) and \(\frac{x}{-2}+\frac{y}{1}=-1\) (C) \(\frac{x}{2}+\frac{y}{3}=1\) and \(\frac{x}{2}+\frac{y}{1}=1\) (D) \(\frac{x}{2}-\frac{y}{3}=1\) and \(\frac{x}{-2}+\frac{y}{1}=1\)

Step-by-Step Solution

Answer

The equation of the line is \(\frac{x}{-2} + \frac{y}{1} = -1\) (Option B).

Step 1: Understanding intercept form

The intercept form of the equation of a line is given by \(\frac{x}{a} + \frac{y}{b} = 1\), where \(a\) is the x-intercept and \(b\) is the y-intercept. To solve the problem, we need to determine \(a\) and \(b\) such that their sum is \(-1\).

Step 2: Setting up the equation based on intercepts

Let \(a\) represent the x-intercept and \(b\) the y-intercept. The problem states that \(a + b = -1\). Thus, the equation of the line becomes \(\frac{x}{a} + \frac{y}{b} = 1\).

Step 3: Finding values for \(a\) and \(b\)

We begin by acknowledging that the given point \((4,3)\) lies on the line. Substituting \(x = 4\) and \(y = 3\) in the equation \(\frac{x}{a} + \frac{y}{b} = 1\), we have \(\frac{4}{a} + \frac{3}{b} = 1\). Additionally, \(a + b = -1\) gives us two equations to solve simultaneously.

Step 4: Solving simultaneous equations

Using \(a + b = -1\) and \(\frac{4}{a} + \frac{3}{b} = 1\), express \(b = -1 - a\) and substitute it into the second equation, leading to \(\frac{4}{a} + \frac{3}{-1-a} = 1\). Solving for \(a\), we find \(a = -2\) and consequently, \(b = 1\).

Step 5: Writing the equation of the line

With \(a = -2\) and \(b = 1\), substitute back into the line equation to get \(\frac{x}{-2} + \frac{y}{1} = 1\). Rearrange if necessary to match the options: \(\frac{x}{-2} + \frac{y}{1} = -1\), which simplifies to \(\frac{x}{-2} + y = -1\).

Step 6: Selecting the correct option

Compare the simplified equation with the given options. Option \(B\), \(\frac{x}{-2} + \frac{y}{1} = -1\), matches our derived equation. Thus, the correct answer is option \(B\).

Key Concepts

Equation of a Line

When we talk about the equation of a line, we are referring to a representation that shows all the points which lie on that line in a coordinate plane. There are different ways to write the equation of a line, but they all serve the same purpose: describing the line's geometry.

In general, the equation of a line in two dimensions is expressed in the standard form, slope-intercept form, or intercept form. The standard form is given as \( Ax + By = C \), where \(A\), \(B\), and \(C\) are real numbers. The slope-intercept form is written as \( y = mx + c \), where \(m\) is the slope and \(c\) is the y-intercept of the line.

A line's equation not only defines the line but allows us to calculate various properties and analyze its relationship with other geometrical entities. By substituting values into a line's equation, we can find coordinates of specific points and verify if a point lies on the line.

Intercept Form

One of the unique ways to represent a line is through its intercept form. This format comes in handy when it is easy to determine where the line crosses the x-axis and y-axis, called intercepts. The intercept form of a line's equation is \( \frac{x}{a} + \frac{y}{b} = 1 \), where \(a\) is the x-intercept and \(b\) is the y-intercept.

• If \(a = 0\), the line is vertical.
• If \(b = 0\), the line is horizontal.

The intercept form is particularly useful when dealing with problems where these intercepts either have specific values or specific relationships, such as their sum, which we encountered in the given exercise.

Once we identify the values of \(a\) and \(b\), substituting them back into the intercept form reveals the specific equation of the line. This equation is valuable for graphing the line and understanding its geometric properties.

Simultaneous Equations

Simultaneous equations are sets of equations with multiple variables that are solved together, as each equation shares some of these variables. Solving such systems means finding a common solution that satisfies all of the equations involved.

In our exercise, we encountered two simultaneous equations, which were:

• \(a + b = -1\)
• \(\frac{4}{a} + \frac{3}{b} = 1\)

To solve these equations, you need to express one variable in terms of another using one equation before substituting into the second equation. In our case, we wrote \(b = -1 - a\) and replaced \(b\) in the second equation.

Solving the transformed equation allows for finding specific values for \(a\) and \(b\). Once these values are obtained, they can be plugged into the original equations to verify they satisfy all conditions. Successfully solving simultaneous equations provides us with essential variables that lead to making sense of more complex geometrical constructs, such as forming precise line equations.