Problem 4 - Calculus for JEE ADVANCED

Question

The general equation of all non-horizontal lines in xy-plane is \(a x+b y=1\), where \(a \neq 0\)

Step-by-Step Solution

Answer

The equation represents a line with slope \(-\frac{a}{b}\) and y-intercept \(\frac{1}{b}\).

Step 1: Understanding the Equation

The given equation is \(ax + by = 1\), where \(a eq 0\). This represents a non-horizontal line because \(a\) is not zero, implying that the line has a slope and is not parallel to the x-axis.

Step 2: Solve for y

Rearrange the general equation \(ax + by = 1\) to solve for \(y\). We get: \[ by = 1 - ax \] Now, divide every term by \(b\) to isolate \(y\): \[ y = \frac{1}{b} - \frac{a}{b}x \]

Step 3: Identify the Slope and Y-intercept

The equation \(y = -\frac{a}{b}x + \frac{1}{b}\) is now in the slope-intercept form \(y = mx + c\), where \(m = -\frac{a}{b}\) and \(c = \frac{1}{b}\). Thus, the slope of the line is \(-\frac{a}{b}\) and the y-intercept is \(\frac{1}{b}\).

Key Concepts

Slope-Intercept Form

When discussing equations of lines, one of the most significant forms is the slope-intercept form. This form is represented as \(y = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept, or the point where the line crosses the y-axis.
This form is particularly useful because it immediately provides information about the direction and steepness of the line, as well as its intersection with the vertical axis.

• **Slope (\(m\)):** Indicates how steep the line is. A positive slope means the line goes upward as it moves from left to right, while a negative slope goes downward.
• **Y-intercept (\(c\)):** This is the value of \(y\) when \(x = 0\), showing where the line crosses the y-axis.

This powerful form simplifies the graphing of linear equations, as it instantly gives us a starting point and direction for sketching the line.

Non-Horizontal Lines

A non-horizontal line in the xy-plane has a noticeable characteristic: it is inclined either upwards or downwards instead of lying flat. Such lines are not parallel to the x-axis, meaning they have a slope other than zero.

Any line defined by the equation \(ax + by = 1\) where \(a eq 0\), cannot be horizontal because the presence of \(x\) with a non-zero coefficient \(a\) ensures that the line inclines. The slope equation \(m = -\frac{a}{b}\) (derived from rearranging and obtaining \(y = mx + c\)) shows that as long as \(a\) is not zero, the line has a specific degree of inclination due to the slope value \(-\frac{a}{b}\).

Without \(a\), we'd have only a vertical line defined by \(b\), which is an entirely different orientation in the plane. Thus, while horizontal lines require different considerations, most of the time you'll encounter these non-horizontal varieties in linear equations.

Slope and Intercept Calculation

Calculating the slope and intercept of a line is a fundamental skill in algebra. With the equation \(y = -\frac{a}{b}x + \frac{1}{b}\) obtained from rearranging the equation \(ax + by = 1\), we identify the slope \(m\) and y-intercept \(c\) directly.

• **Slope (\(m\)):** This is calculated as \(-\frac{a}{b}\). It reflects how much the \(y\)-value changes for a unit change in \(x\). A negative slope implies the line descends as \(x\) increases.
• **Y-intercept (\(c\)):** The y-intercept here is \(\frac{1}{b}\). It tells us where the line meets the y-axis, which is when \(x = 0\).

These calculations are straightforward yet insightful, as they immediately allow you to visualize the line's behavior and position in the coordinate system. This understanding is crucial, not just for solving problems, but for graphically interpreting linear relationships in mathematics.