Definitions [6]
The slope m of a line is m = tanθ
where θ is the inclination of the line with the positive x-axis.
An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.
Find the equation of the line which passes through the point (3, 4) and is such that the portion of it intercepted between the axes is divided by the point in the ratio 2:3.
The equation of the line with intercepts a and b is \[\frac{x}{a} + \frac{y}{b} = 1\] .Since the line meets the coordinate axes at A and B, the coordinates are A (a, 0) and B (0, b).
Let the given point be P (3, 4).
Here,
\[AP : BP = 2 : 3\]
\[\therefore 3 = \frac{2 \times 0 + 3 \times a}{2 + 3}, 4 = \frac{2 \times b + 3 \times 0}{2 + 3}\]
\[ \Rightarrow 3a = 15, 2b = 20\]
\[ \Rightarrow a = 5, b = 10\]
Hence, the equation of the line is
\[\frac{x}{5} + \frac{y}{10} = 1\]
\[ \Rightarrow 2x + y = 10\]
The two mutually perpendicular number lines intersecting each other at their zeroes are called rectangular axes or coordinate axes, or axes of reference.
The position of a point in a plane is expressed by a pair of numbers, one concerning the x-axis and the other concerning the y-axis. called co-ordinates.
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x → distance from y-axis (abscissa)
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y → distance from x-axis (ordinate)
Locus is the path traced by a moving point, which moves so as to satisfy a certain given condition/conditions.
Formulae [4]
\[m=\frac{y_2-y_1}{x_2-x_1}\]
When slope and y-intercept are given
y = mx + c
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m = slope
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c = y-intercept (value of y when x = 0)
When two points are given
\[\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}\]
When the slope and one point are given
y − y1 = m(x − x1)
Key Points
Nature of Slope
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m > 0 → rising line
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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
Sign Convention
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Right of y-axis → +x
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Left of y-axis → −x
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Above x-axis → +y
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Below x-axis → −y
Standard Line Results
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x = 0 → y-axis
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y = 0 → x-axis
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x = a → line parallel to the y-axis
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y = b → line parallel to the x-axis
Quadrant Reminder
| Quadrant | Sign of (x, y) |
|---|---|
| I | (+, +) |
| II | (−, +) |
| III | (−, −) |
| IV | (+, −) |
Concepts [11]
- Brief Recall of Two Dimensional Geometry from Earlier Classes
- Shifting of Origin
- Concept of Slope (or, gradient)
- Various Forms of the Equation of a Line
- Equations of Line in Different Forms
- Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
- Distance of a Point from a Line
- Equations of Bisectors of Angle Between Two Lines
- Family & Concurrent Lines
- Co-ordinate Geometry
- Locus
