Definitions [4]
An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.
The angle of inclination, or simply the inclination of a line, is the angle θ that the part of the line above the x-axis makes with the positive direction of the x-axis and is measured in an anticlockwise direction.
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Anticlockwise → Positive inclination
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Clockwise → Negative inclination
The slope m of a line is m = tanθ
where θ is the inclination of the line with the positive x-axis.
x-intercept: Point where a line cuts the x-axis, y = 0
y-intercept: Point where a line cuts the y-axis, x = 0
Formulae [4]
From general form:
- Slope (m) = −a / b
- Y-intercept = −c / b
\[m=\frac{y_2-y_1}{x_2-x_1}\]
For line ax + by + c = 0
x-intercept:
\[\left(-\frac{c}{a},0\right)\]
y-intercept:
\[\left(0,-\frac{c}{b}\right)\]
One line has a slope m = tanθ
The other equally inclined line has a slope m = − tanθ
Slopes are equal in magnitude, opposite in sign
Key Points
| Form | Formula |
|---|---|
| X-axis | y = 0 |
| Y-axis | x = 0 |
| Parallel to the X-axis | y = b or y = -b |
| Parallel to the Y-axis | x = a or x = -a |
| Slope-point form | y − y₁ = m(x − x₁) |
| Two-point form | \[\frac{y-y_{1}}{y_{1}-y_{2}}=\frac{x-x_{1}}{x_{1}-x_{2}}\] |
| Slope-intercept form | y = mx + c |
| Intercept form | \[\frac{x}{\mathrm{a}}+\frac{y}{\mathrm{b}}=1\] |
| Normal form | x cosα + y sinα = p |
| Parametric form | \[\frac{x-x_{1}}{\cos\theta}=\frac{y-y_{1}}{\sin\theta}=r\] |
Position of a Point:
For line: ax₁ + by₁ + c
- If ax₁ + by₁ + c = 0 → Point lies on the line
- If ax₁ + by₁ + c < 0 → Point lies on one side (origin side)
- If ax₁ + by₁ + c > 0 → Point lies on other side
Standard Results
-
x-axis / parallel to x-axis → θ = 0∘
-
y-axis / parallel to y-axis → θ=90∘
Line Types
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Horizontal line → parallel to x-axis
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Vertical line → parallel to y-axis
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Oblique line → neither parallel to the x-axis nor the y-axis
Nature of Slope
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m > 0 → rising line
-
m < 0 → falling line
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m = 0 → horizontal line
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m = ∞→ vertical line
Parallel Lines
Two lines are parallel ⇔ , their slopes are equal, m1 = m2
Perpendicular Lines
Two lines are perpendicular ⇔
Collinearity of Three Points
Points A, B, and C are collinear
Method 1: Distance method
AB + BC = AC
Method 2: Slope method
Slope of AB = Slope of BC
-
x-intercept:
Right of origin → positive
Left of origin → negative -
y-intercept:
Above origin → positive
Below origin → negative
