Straight Line
Easy Overview
The straight line is the simplest geometric shape there is. Two points, infinite length, constant direction. But simplicity doesn't mean lack of depth. A straight line can be described in half a dozen different forms, each useful in different contexts. The relationship between lines — whether they're parallel, perpendicular, or intersecting — tells you something about the space they occupy. In coordinate geometry, every line has an equation. The most intuitive is y = mx + c, where m is the slope (steepness) and c is the y-intercept (where the line crosses the y-axis). But a line can also be written in point-slope form (y − y₁ = m(x − x₁)), two-point form, intercept form (x/a + y/b = 1), general form (ax + by + c = 0), normal form, and parametric form. Why so many? Because different problems give you different information. If you have a point and a slope, use point-slope. If you have two intercepts, use intercept form. If you want to find distances, use the general form. Knowing which form to pick is half the battle. The slope itself carries a lot of meaning. A positive slope means the line goes up as you move right. A negative slope means it goes down. A zero slope is horizontal. An undefined slope (the denominator is zero) means the line is vertical. The slope between any two points on the line is constant — that's what makes it a straight line. When you have two lines, interesting things happen. Parallel lines have equal slopes — they never meet, like railway tracks. Perpendicular lines have slopes that multiply to −1 — they meet at a right angle. The angle between two lines can be found using tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. Distance formulas are another key part of this chapter. The perpendicular distance from a point to a line is |ax₁ + by₁ + c|/√(a² + b²). The distance between two parallel lines is |c₂ − c₁|/√(a² + b²). These formulas are essential for geometry problems and for understanding the concept of 'closest distance'. The final major topic is the family of lines. A family of lines is a set of lines sharing a common property — all lines passing through a given point, or all lines with a given slope. The equation L₁ + λL₂ = 0 represents all lines passing through the intersection of L₁ = 0 and L₂ = 0. By varying λ, you get different lines, but they all pass through the same intersection point. By the end, you'll be able to find the equation of any line given its geometric description, determine relationships between lines, and solve distance and intersection problems. The straight line is the building block of coordinate geometry — master it, and curves become much easier.
Slope or gradient of a line — the measure of steepness
The slope m of a line is the ratio of vertical change to horizontal change. Given two points (x₁, y₁) and (x₂, y₂), m = (y₂ − y₁)/(x₂ − x₁) = rise/run. A positive slope tilts upward to the right, negative tilts downward, zero is horizontal, and undefined (division by zero) means vertical. The larger the absolute value of m, the steeper the line.
Angle between two lines
If two lines have slopes m₁ and m₂, the acute angle θ between them is given by tan θ = |(m₂ − m₁)/(1 + m₁m₂)|. If the denominator 1 + m₁m₂ = 0, then θ = 90° — the lines are perpendicular. This formula is derived from the fact that the angle a line makes with the x-axis is tan⁻¹(m), and the difference of two such angles gives the angle between them.
Condition for parallel lines
Two lines are parallel if and only if their slopes are equal: m₁ = m₂. For lines in general form ax + by + c = 0, parallel lines have the same a and b coefficients but different c. So lines 2x + 3y − 5 = 0 and 2x + 3y + 7 = 0 are parallel. Visually, parallel lines never intersect and maintain a constant distance between them.
Condition for perpendicular lines
Two lines are perpendicular if and only if the product of their slopes is −1: m₁·m₂ = −1. Equivalently, m₂ = −1/m₁ — one slope is the negative reciprocal of the other. For example, if a line has slope 2, any perpendicular line has slope −½. For lines in general form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition is a₁a₂ + b₁b₂ = 0.
Slope-intercept form — the most familiar
y = mx + c, where m is the slope and c is the y-intercept (the y-coordinate where x = 0). This is the most intuitive form because you can immediately see the slope and where the line crosses the y-axis. To graph a line in this form, start at (0, c) and use the slope m = rise/run to find other points. The x-intercept is found by setting y = 0: x = −c/m.
Point-slope form — given a point and a slope
y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a known point on the line. This is the form to use when you're given one point and the slope. It's derived directly from the slope formula. From here you can easily expand to slope-intercept or general form. For example, a line through (2, 3) with slope 4 is y − 3 = 4(x − 2).
Two-point form — when you know two points
(y − y₁)/(x − x₁) = (y₂ − y₁)/(x₂ − x₁). This directly expresses the constancy of slope between any two points on the line. Given (1, 2) and (3, 6), the slope is (6−2)/(3−1) = 2, so the equation becomes (y − 2)/(x − 1) = 2, or y = 2x. This form is especially useful when neither intercept nor slope is obvious.
Double-intercept form — intercepts at a glance
x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. This form is elegant because the intercepts are right there in the equation. The line crosses the x-axis at (a, 0) and the y-axis at (0, b). It's the easiest form to work with when you know where a line crosses the axes. For example, a line with x-intercept 4 and y-intercept 3 is x/4 + y/3 = 1.
Normal form — using the perpendicular distance
x cos α + y sin α = p, where p is the perpendicular distance from the origin to the line, and α is the angle that this perpendicular makes with the positive x-axis. This form is particularly useful when you know the line's distance from the origin and its orientation. The parameters p and α uniquely define the line.
General equation — the universal form
ax + by + c = 0, where a, b, and c are constants, and a and b are not both zero. Every line can be written in this form. The slope is −a/b (if b ≠ 0), the x-intercept is −c/a (if a ≠ 0), and the y-intercept is −c/b (if b ≠ 0). This is the 'parent' form from which others can be derived. It's also the most convenient form for distance calculations.
Reduction of general form to other forms
Given ax + by + c = 0, you can convert to slope-intercept: y = (−a/b)x + (−c/b). To intercept form: x/(−c/a) + y/(−c/b) = 1. To normal form: divide by √(a² + b²) and adjust signs so that the constant term becomes −p. Being able to switch between forms is a key skill — it lets you extract whatever information you need from any given form.
Distance from a point to a line
The perpendicular distance from point (x₁, y₁) to line ax + by + c = 0 is d = |ax₁ + by₁ + c| / √(a² + b²). The absolute value ensures distance is positive. The numerator is the absolute value of the line equation evaluated at the point — it represents how 'far' the point is from satisfying the equation. The denominator is the normalisation factor.
Distance between two parallel lines
For parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0, the distance between them is d = |c₂ − c₁| / √(a² + b²). Note that the a and b coefficients must be the same before applying this formula. If they're not, multiply through to make them match. For example, 2x + 3y − 5 = 0 and 2x + 3y + 7 = 0 are apart by |7 − (−5)|/√(13) = 12/√13 units.
Family of lines passing through the intersection of two lines
If two lines L₁: a₁x + b₁y + c₁ = 0 and L₂: a₂x + b₂y + c₂ = 0 intersect, then L₁ + λL₂ = 0 (i.e., (a₁x + b₁y + c₁) + λ(a₂x + b₂y + c₂) = 0) represents all lines passing through their intersection point. Each value of λ (a real parameter) gives a different line, except L₂ itself. This is a powerful technique for finding a specific line through the intersection that also satisfies another condition.
Equation of a line in parametric form
A line can be represented parametrically as x = x₁ + t·cos θ, y = y₁ + t·sin θ, where (x₁, y₁) is a point on the line, θ is the direction angle of the line (so slope m = tan θ), and t is the parameter (distance from the point along the line). This form is particularly useful in vector geometry and for describing motion along a straight line path.
Collinearity of three points
Three points A, B, C are collinear if they lie on the same straight line. This can be checked in multiple ways: (1) slope of AB = slope of BC, (2) area of triangle ABC = 0 (using determinant formula), (3) the distance AB + BC = AC. These conditions are equivalent and useful for different problem types. The determinant method is particularly clean: |x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1| = 0.
Locus of a moving point
A locus is the set of all points satisfying a given condition. For straight lines, common loci include: a point that moves so that its distance from a fixed point is constant (a circle), or a point that's equidistant from two fixed points (the perpendicular bisector — a straight line). Finding the equation of a locus involves translating geometric conditions into algebraic equations.
Combined equation of a pair of lines
A second-degree homogeneous equation like ax² + 2hxy + by² = 0 represents a pair of lines passing through the origin if h² ≥ ab. Each individual line's slope satisfies the quadratic bm² + 2hm + a = 0. The angle between the two lines is given by tan θ = 2√(h² − ab)/|a + b|. This connects the straight line chapter with the quadratic equations chapter.
Key Points
- •Slope m = (y₂ − y₁)/(x₂ − x₁) = rise/run
- •tan θ = |(m₂ − m₁)/(1 + m₁m₂)| for the angle between two lines
- •Parallel lines: m₁ = m₂
- •Perpendicular lines: m₁·m₂ = −1
- •y = mx + c — slope-intercept form
- •y − y₁ = m(x − x₁) — point-slope form
- •(y − y₁)/(x − x₁) = (y₂ − y₁)/(x₂ − x₁) — two-point form
- •x/a + y/b = 1 — double-intercept form
- •x cos α + y sin α = p — normal form
- •ax + by + c = 0 — general form (covers all lines)
- •Distance from point to line: |ax₁ + by₁ + c|/√(a² + b²)
- •Distance between parallel lines: |c₂ − c₁|/√(a² + b²)
- •Family of lines through intersection: L₁ + λL₂ = 0
- •Three points are collinear if slope AB = slope BC (or area = 0)
- •For pair of lines: ax² + 2hxy + by² = 0 represents two lines if h² ≥ ab
Practice Questions
- Find the equation of a line passing through (2, 3) with slope −4 in all five forms.
- Find the distance between the parallel lines 3x + 4y − 7 = 0 and 3x + 4y + 8 = 0.
- Find the equation of a line perpendicular to 2x − 3y + 5 = 0 and passing through the point (1, −1).
- Show that points A(1, 2), B(3, 6), and C(5, 10) are collinear using two different methods.
- A line makes positive intercepts on the axes whose sum is 7 and product is 12. Find its equation.
- Find the equation of the line passing through the intersection of 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
- Find the angle between the lines y = 2x + 3 and 3x + y − 4 = 0.
- A point moves so that its distance from the point (1, 2) is always twice its distance from the line 3x + 4y − 5 = 0. Find the equation of its locus.