Unit 7: Straight Lines
Contents
1. Slope of a Line
The Slope (or gradient) of a line is a measure of its steepness and direction. It is denoted by m.
- Using Inclination: If θ is the angle a line makes with the positive x-axis, then m = tan θ.
- Using Two Points: If a line passes through (x1, y1) and (x2, y2), then: m = (y2 - y1) / (x2 - x1)
Conditions: Parallel lines have equal slopes (m1 = m2). Perpendicular lines have a product of slopes equal to -1 (m1 · m2 = -1).
2. Angle Between Two Lines
When two lines with slopes m1 and m2 intersect, the acute angle α between them is given by:
3. Various Forms of Equations of a Line
Depending on the information available, we use different forms to represent a straight line in the Cartesian plane.
| Form Name | Equation | Given Information |
|---|---|---|
| Slope-Intercept Form | y = mx + c | Slope (m) and y-intercept (c). |
| Point-Slope Form | y - y1 = m(x - x1) | Slope (m) and a point (x1, y1). |
| Two-Point Form | y - y1 = (y2 - y1) / (x2 - x1)(x - x1) | Two points (x1, y1) and (x2, y2). |
| Intercept Form | (x) / (a) + (y) / (b) = 1 | x-intercept (a) and y-intercept (b). |
4. Distance of a Point from a Line
The perpendicular distance d from a point (x1, y1) to a line Ax + By + C = 0 is calculated as:
Distance between two parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0 is:
5. Exam Focus Enhancements
- The Intercept Trick: If you are asked to find the intercepts of 3x + 4y = 12, simply divide the whole equation by 12 to get (x) / (4) + (y) / (3) = 1. The intercepts are (4, 0) and (0, 3).
- Collinearity: Three points are collinear if the slope of AB equals the slope of BC.
- General Form: Always try to leave your final answer in the general form Ax + By + C = 0.
- Vertical Line Slope: Remember that the slope of a vertical line is undefined (tan 90°).
- Sign Errors: When using the distance formula, ensure all terms are on one side of the equation (= 0) before identifying A, B, and C.
- Intercept Form: The intercept form must have a "+1" on the right side. If it's -1, multiply the whole equation by -1.
Q: How do I find the slope of a line given in general form Ax + By + C = 0?
A: Convert it to y = mx + c form. The slope is m = -A/B.
Q: What is the slope of the x-axis and y-axis?
A: The x-axis has a slope of 0 (tan 0°). The y-axis has an undefined slope (tan 90°).