Straight Lines — Study Notes

1. Introduction

Coordinate geometry fuses algebra with geometry by assigning numbers to points and translating geometric figures into equations. This chapter develops the analytic study of straight lines: their slopes, their various equations, the angles between them, and the distances associated with them. It is the foundation of the analytic geometry of the plane.

The chapter begins with the distance formula and the section formula, then introduces the concept of slope of a line and its relation to angle of inclination. It develops several forms of the equation of a straight line: point-slope, slope-intercept, two-point, intercept, and normal forms. It then studies parallelism, perpendicularity, the angle between two lines, and the distance of a point from a line.

Straight lines are the simplest curves and appear everywhere in applications: linear models in physics and economics, road and railway geometry, and the graphical solution of equations. The concepts developed here are prerequisites for conic sections, and the general techniques of coordinate geometry will be used throughout higher mathematics.

2. Coordinate Geometry Basics

Distance Between Two Points

The distance between two points P(x1, y1) and Q(x2, y2) is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Section Formula

If the point R divides the line segment joining P(x1, y1) and Q(x2, y2) in the ratio m : n, then:

R = ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n))

If the ratio is taken internally. For the midpoint, m : n = 1 : 1, giving:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

3. Slope of a Line

The slope (or gradient) of a line is the measure of its steepness. If the line makes an angle theta with the positive direction of the x-axis, then:

m = tan theta

For a line passing through P(x1, y1) and Q(x2, y2), the slope is:

m = (y2 - y1)/(x2 - x1)

Properties of Slope

• A horizontal line has slope 0.
• A vertical line has undefined slope (division by zero).
• The slope is positive when the line rises from left to right and negative when it falls.
• Two parallel lines have equal slopes.
• Two perpendicular lines with slopes m1 and m2 satisfy m1 m2 = -1.

4. Various Forms of the Equation of a Line

Point-Slope Form

A line through a point P(x1, y1) with slope m has equation:

y - y1 = m(x - x1)

Slope-Intercept Form

A line with slope m and y-intercept c has equation:

y = mx + c

Two-Point Form

A line through points (x1, y1) and (x2, y2) has equation:

(y - y1)/(y2 - y1) = (x - x1)/(x2 - x1)

Intercept Form

A line making intercepts a on the x-axis and b on the y-axis has equation:

x/a + y/b = 1

Normal Form

If p is the length of the perpendicular from the origin to the line and the perpendicular makes an angle alpha with the positive x-axis, then:

x cos alpha + y sin alpha = p

General Equation

The general linear equation is Ax + By + C = 0, where A and B are not both zero. Its slope is -A/B.

5. Angle Between Two Lines

If two lines have slopes m1 and m2, the tangent of the angle theta between them is:

tan theta = |(m2 - m1)/(1 + m1 m2)|

From this formula:

• If m1 = m2, the lines are parallel (theta = 0).
• If m1 m2 = -1, the lines are perpendicular (theta = 90 degrees).

Condition for Parallel and Perpendicular

Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1.

6. Distance of a Point From a Line

The perpendicular distance of a point P(x1, y1) from the line Ax + By + C = 0 is:

d = |Ax1 + By1 + C|/sqrt(A^2 + B^2)

Distance Between Two Parallel Lines

If two parallel lines are Ax + By + C1 = 0 and Ax + By + C2 = 0, the distance between them is:

d = |C1 - C2|/sqrt(A^2 + B^2)

Quick Revision Tables

Table 1: Forms of the Equation of a Straight Line

Table 2: Key Formulas and Conditions

Mind Map

Important Diagrams (SVG)

Diagram 1: Slope of a Line and Angle of Inclination

Diagram 2: Distance of a Point from a Line

Common Mistakes

1. Using the wrong formula for slope. The slope is (y2 - y1)/(x2 - x1), not (x2 - x1)/(y2 - y1).
2. Saying a vertical line has slope 0. A vertical line has undefined slope because the denominator x2 - x1 is zero.
3. Confusing the perpendicular condition. Perpendicular lines satisfy m1 m2 = -1, not m1 = -m2.
4. In the section formula, misplacing the ratio m : n. The coordinate of R is (m x2 + n x1)/(m + n) for internal division in ratio m : n.
5. Forgetting the absolute value in the distance formula. Distance is never negative.
6. Using the slope-intercept form y = mx + c when the line is vertical; vertical lines cannot be written in this form.
7. When converting a line to normal form, forgetting that both sides must be divided by the same factor so that the equation remains x cos alpha + y sin alpha = p.
8. Mixing up intercept form: x/a + y/b = 1, where a and b are the x and y intercepts, not the coordinates of a point.
9. Cancelling the denominator (1 + m1 m2) in the angle formula without checking that the lines are not perpendicular, where the formula becomes undefined.

Exam Tips

1. To find the slope quickly, rewrite the general equation Ax + By + C = 0 as y = (-A/B)x - C/B; the slope is -A/B.
2. When two lines are parallel, keep the coefficients of x and y the same and change only the constant term.
3. For perpendicular lines, swap the coefficients of x and y and change the sign of one of them; the slopes then multiply to -1.
4. In distance problems, always present the formula with the absolute value and compute the denominator sqrt(A^2 + B^2) carefully.
5. To decide between forms, read what information is given: slope and a point point to point-slope form; two intercepts point to intercept form.
6. For the angle between lines, first check parallelism and perpendicularity, which are special cases of the general angle formula.
7. Practise problems where the line passes through the intersection of two given lines, since they combine the section and line equations.

Conclusion

Straight lines provide the first and simplest bridge between algebra and geometry. The slope quantifies direction, the various equation forms capture lines from different viewpoints, and the conditions for parallelism and perpendicularity make it easy to reason about pairs of lines. Distance formulas extend analytic geometry to quantitative problems involving points and lines. These tools are not only examined directly but are also the foundation for conic sections, linear programming, and the study of curves in general. Mastery of coordinate geometry of straight lines gives students a powerful and general method for solving geometric problems algebraically.