Definitions [4]
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)
A Pictograph is a chart that uses pictures or symbols to represent data. Each picture stands for a specific number of items, making the data easy to understand at a glance.
An equation of the form ax + by + c = 0 represents a straight line and is known as a linear equation.
Formulae [2]
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
From general form:
- Slope (m) = −a / b
- Y-intercept = −c / b
Theorems and Laws [1]
If the points p (x, y) is point equidistant from the points A (5, 1)and B (–1, 5), Prove that 3x = 2y
As per the question, we have
AP = BP
`⇒ sqrt((x -5)^2 +(y-1)^2) = sqrt((x+1)^2 +(y-5)^2)`
`⇒(x-5)^2 +(y-1)^2 = (x+1)^2 +(y-5)^2` (Squaring both sides)
`⇒x^2 - 10x +25 + y^2 -2y +1 = x^2 +2x +1+y^2 -10y+25`
⇒ –10x – 2y = 2x – 10y
⇒ 8y = 12x
⇒ 3x = 2y
Key Points
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 | (+, −) |
| 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
