Definitions [2]
Definition: Co-ordinate Axes
The two mutually perpendicular number lines intersecting each other at their zeroes are called rectangular axes or coordinate axes, or axes of reference.
Definition: Co-ordinates
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)
Formulae [1]
Formula: Foundation of Coordinate Geometry
| Sr. No. | Name | Condition | Formula |
|---|---|---|---|
| i. | Distance Formula | Two points P(x₁, y₁), Q(x₂, y₂) | \[\mathrm{d(PQ)}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\] |
| a. | Internal Division | P divides AB in the ratio m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+ny_{1}}{m+n}\right)}\] |
| b. | Midpoint Formula | P is the midpoint of AB | \[\mathrm{P}\equiv\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)\] |
| c. | External Division | P divides AB externally in m: n | \[\mathrm{P\equiv\left(\frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-ny_{1}}{m-n}\right)}\] |
| iii. | Centroid Formula | Triangle with vertices A, B, C | \[\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)\] |
Key Points
Key Points: Co-ordinate Geometry
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 | (+, −) |
