Problem 42 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the \(x\)-intercept of some line \(L\) is double as that of the line, \(3 x+4 y=12\) and the \(y\)-intercept of \(L\) is half as that of the same line, then the slope of \(L\) is : [Online April 22, 2013] (a) \(-3\) (b) \(-3 / 8\) (c) \(-3 / 2\) (d) \(-3 / 16\)

Step-by-Step Solution

Answer

The slope of line \(L\) is \(-\frac{3}{16}\), option (d).

Step 1: Determine x-intercept of Given Line

The equation of the given line is \(3x + 4y = 12\). To find the \(x\)-intercept, set \(y = 0\) and solve for \(x\). \[3x + 4(0) = 12\] \[3x = 12\] \[x = 4\]. So, the \(x\)-intercept is 4.

Step 2: Determine y-intercept of Given Line

Use the line equation \(3x + 4y = 12\) and set \(x = 0\) to solve for \(y\). \[3(0) + 4y = 12\] \[4y = 12\] \[y = 3\]. Thus, the \(y\)-intercept is 3.

Step 3: Calculate Intercepts of Line L

Given that the \(x\)-intercept of \(L\) is double that of the given line, calculate \(2 \times 4 = 8\). The \(y\)-intercept of \(L\) is half that of the given line, so \(\frac{1}{2} \times 3 = \frac{3}{2}\).

Step 4: Find Slope of Line L

For line \(L\), the equation can be represented in the form \(y = mx + c\). When expressed using the given intercepts, its slope \(m\) is calculated as \(m = \frac{\text{change in } y}{\text{change in } x} = \frac{\frac{3}{2} - 0}{0 - 8}\). \[m = \frac{\frac{3}{2}}{-8} = -\frac{3}{16}\].

Step 5: Select Correct Answer

The slope of the line \(L\) is \(-\frac{3}{16}\). The correct option from the provided choices is (d).

Key Concepts

Slope of a Line

In coordinate geometry, the slope is a measure that describes the steepness and direction of a line. It is typically represented by the letter "\(m\)" in the equation of a line. When calculating the slope between two points \( (x_1, y_1) \) and \((x_2, y_2)\), the formula is:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

The slope can be positive, negative, zero, or undefined:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope means the line is perfectly horizontal.
• An undefined slope means the line is perfectly vertical.

For line \(L\) in our problem, the intercepts helped us determine the slope. The vertical change was \(\frac{3}{2}\), and the horizontal change was \(-8\), leading to a slope of \(-\frac{3}{16}\). This indicates that line \(L\) slopes downward as it moves from left to right.

Intercepts

Intercepts are points where a line crosses the axes in a coordinate plane.
There are two types of intercepts:

• The \(x\)-intercept, which is where the line crosses the x-axis. This happens when \(y = 0\).
• The \(y\)-intercept, which is where the line crosses the y-axis. This is found when \(x = 0\).

For the line \(3x + 4y = 12\):

• The \(x\)-intercept is found by setting \(y=0\), solving to get \((4, 0)\).
• The \(y\)-intercept is found by setting \(x=0\), giving us \((0, 3)\).

The line \(L\) modifies these values as its intercepts:

• \(x\)-intercept: \(8\) (double that of the given line)
• \(y\)-intercept: \(\frac{3}{2}\) (half of the given line)

These intercepts helped form the basis for finding the slope of line \(L\).

Linear Equations

Linear equations in two variables describe a straight line in the coordinate plane.
They can generally be written in the form \(Ax + By = C\) or \(y = mx + c\), where:

• \(A\), \(B\), and \(C\) are constants
• \(m\) is the slope
• \(c\) is the \(y\)-intercept

The standard form, \(Ax + By = C\), is useful for finding intercepts as we can set \(x = 0\) or \(y = 0\).
The slope-intercept form, \(y = mx + c\), directly shows the slope and \(y\)-intercept, making it easy to graph the line.
For our problem's line \(L\), using intercepts derived from the given line \(3x + 4y = 12\), the equation took the form \(y = mx + \frac{3}{2}\).
From this form, we identified the slope \(-\frac{3}{16}\) after performing calculations based on the changes between intercepts.