Coordinate Geometry – Class 10 (CBSE)
Why this topic?
Coordinate geometry turns geometric problems into algebraic ones. All the concepts you learn here are reused in higher‑level math, physics, engineering and many competitive exams (IIT‑JEE, NPTEL, etc.). Mastering the derivations and tricky points will give you a solid base for those exams.
1. The Cartesian Plane
| Symbol | Meaning |
|---|---|
(x , y) |
A point in the plane |
x‑axis |
Horizontal axis (positive to the right) |
y‑axis |
Vertical axis (positive upwards) |
Origin O |
(0, 0) |
| Quadrants | I (+,+), II (−,+), III (−,−), IV (+,−) |
Note: The first coordinate is the abscissa (x‑coordinate), the second is the ordinate (y‑coordinate).
2. Distance Between Two Points
Formula
[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} ]
Derivation
-
Consider the right‑angled triangle formed by the two points and the perpendiculars to the axes.
-
By Pythagoras:
[ d^2 = (x_2-x_1)^2 + (y_2-y_1)^2 ]
-
Take the square root to get (d).
Key points
- Works for any two points, even if they lie in different quadrants.
- If (x_1=x_2) → vertical line → distance is (|y_2-y_1|).
- If (y_1=y_2) → horizontal line → distance is (|x_2-x_1|).
3. Mid‑Point of a Segment
Formula
[ M!\left(\frac{x_1+x_2}{2},; \frac{y_1+y_2}{2}\right) ]
Derivation
Average the coordinates of the endpoints; the midpoint is exactly halfway along the segment.
4. Slope of a Line
Definition
[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1} ]
Derivation
- From two points (P(x_1,y_1)) and (Q(x_2,y_2)), the change in y over the change in x gives the steepness.
- For a horizontal line, (y_2=y_1) → (m=0).
- For a vertical line, (x_2=x_1) → slope is undefined (or “infinite”).
5. Equations of a Straight Line
| Form | General Equation | When to use |
|---|---|---|
| Point‑Slope | (y-y_1 = m(x-x_1)) | Given a point and slope |
| Slope‑Intercept | (y = mx + c) | Given slope & y‑intercept (c) |
| General | (Ax + By + C = 0) | For algebraic manipulation |
| Intercept | (\frac{x}{a} + \frac{y}{b} = 1) | Given x‑ and y‑intercepts ((a,0)) and ((0,b)) |
Derivations
-
Point‑Slope
From two points (P(x_1,y_1)) and (Q(x_2,y_2)):[ m = \frac{y_2-y_1}{x_2-x_1} \quad\Rightarrow\quad y-y_1 = m(x-x_1) ]
-
Slope‑Intercept
Let (c) be the y‑intercept (value of y when (x=0)). The line’s slope is (m). -
General
Multiply the slope‑intercept by a common denominator to clear fractions; bring all terms to the left. -
Intercept
If a line cuts the axes at ((a,0)) and ((0,b)), then:[ \frac{x}{a} + \frac{y}{b} = 1 ]
(Derives from solving for the line passing through those two points.)
6. Parallel & Perpendicular Lines
| Property | Parallel | Perpendicular |
|---|---|---|
| Slopes | Same ((m_1=m_2)) | Product (-1) ((m_1m_2=-1)) |
| Condition for vertical/horizontal | If one is vertical, the other also vertical. | If one is vertical, the other horizontal. |
Example
Given line (y = 3x + 5):
- Parallel line: (y = 3x + c) (any (c)).
- Perpendicular line: (y