Q.Find the equation of the bisector of the angle between the coordinate axes.

The Intuition: What Is a Straight Line?

Before any formula, think about what a straight line does. If you walk from your home to school, the shortest path is a straight line. If you stretch a thread tight between two pins, it forms a straight line. The key property is no turning — the direction never changes.

In coordinate geometry, we describe this mathematically. A straight line is the set of all points $(x, y)$ that satisfy a simple rule: the slope (steepness) between any two points on the line is constant.

The Precise Definition

A straight line in the $xy$-plane is the graph of a linear equation in two variables. The most general form is:

where $A$, $B$, and $C$ are real numbers, and $A$ and $B$ are not both zero.

The Slope: Why It Matters

The slope $m$ measures how steep the line is. It is the ratio of the vertical change to the horizontal change between any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line:

If the line is horizontal, $m = 0$. If it is vertical, the slope is undefined (division by zero). A positive slope means the line rises as you move right; a negative slope means it falls.

Common Forms of the Equation

You will encounter these forms frequently. Each is useful in different situations.

1. Slope-Intercept Form

Here $m$ is the slope and $c$ is the $y$-intercept — the point where the line crosses the $y$-axis. This is the most intuitive form: you see the slope and the starting height immediately.

2. Point-Slope Form

Use this when you know the slope $m$ and one point $(x_1, y_1)$ on the line.

3. Two-Point Form

Use this when you know two distinct points on the line.

4. Intercept Form

Here $a$ is the $x$-intercept and $b$ is the $y$-intercept. This form is convenient when the line cuts the axes at known points.

Special Cases to Watch For

• Line through the origin: $y = mx$ (no constant term).
• Horizontal line: $y = k$ (slope $0$).
• Vertical line: $x = k$ (slope undefined).
• Parallel lines: Two lines are parallel if they have the same slope ($m_1 = m_2$).
• Perpendicular lines: Two lines are perpendicular if the product of their slopes is $-1$ ($m_1 \cdot m_2 = -1$), provided neither is vertical.

Distance Between a Point and a Line

Given a line $Ax + By + C = 0$ and a point $(x_1, y_1)$, the perpendicular distance $d$ is:

This formula is exact and works for any line. The absolute value ensures distance is always positive.

Putting It Together

When you see a problem about a straight line, ask yourself:

1. What information do I have? (slope, one point, two points, intercepts)
2. Which form of the equation fits best?
3. Do I need to find slope, intercepts, distance, or intersection with another line?

The straight line is the simplest curve in coordinate geometry, but it is the foundation for everything that follows — circles, parabolas, and beyond. Master it well.