Coordinate Geometry – Class 10 (CBSE)
Goal: Understand how algebraic equations describe geometric entities (points, lines, circles) in the Cartesian plane. Build a solid foundation that can be extended to higher‑level analytic geometry (MIT, 1st‑year).
Prerequisites: Basic algebra, quadratic equations, Pythagoras’ theorem, properties of real numbers.
1. The Cartesian Coordinate System
| Concept | Description |
|---|---|
| Axes | Two perpendicular lines, traditionally called x‑axis (horizontal) and y‑axis (vertical). |
| Origin | Intersection of the axes, denoted (O(0,0)). |
| Coordinate of a point | Ordered pair ((x, y)) where (x) is the abscissa (horizontal distance from the origin) and (y) is the ordinate (vertical distance). |
| Quadrants | 1st (x>0,y>0), 2nd (x<0,y>0), 3rd (x<0,y<0), 4th (x>0,y<0). |
Key idea: Every point in the plane corresponds bijectively to a unique ordered pair ((x, y)) of real numbers.
2. Distance Between Two Points
From first principles
Let (P(x_1, y_1)) and (Q(x_2, y_2)).
Draw the right triangle (PQR) with (R) projected onto the axes.
[ PQ^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 ]
Hence
[ \boxed{\text{Distance } d(P,Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}} ]
Proof uses Pythagoras’ theorem in (\triangle PQR).
Edge cases
| Situation | Result |
|---|---|
| (P=Q) | (d=0). |
| Horizontal line ((y_1=y_2)) | (d= |
| Vertical line ((x_1=x_2)) | (d= |
3. Midpoint of a Segment
Let (P(x_1, y_1)) and (Q(x_2, y_2)).
The midpoint (M) is
[ \boxed{M!\left(\frac{x_1+x_2}{2},, \frac{y_1+y_2}{2}\right)} ]
Derivation (centroid of a 2‑point set)
- The x‑coordinate of the midpoint is the average of the x‑coordinates, because the horizontal distance is divided equally.
- Similarly for the y‑coordinate.
Edge case – If (P=Q), the midpoint coincides with the point.
4. Slope of a Line
For distinct points (P(x_1, y_1)) and (Q(x_2, y_2)), the slope (m) is
[ \boxed{m = \frac{y_2 - y_1}{x_2 - x_1}} ]
First‑principles reasoning
The slope is the ratio of the rise to the run: vertical change divided by horizontal change.
If (x_1=x_2) the denominator vanishes → vertical line → slope (m) is undefined (∞).
If (y_1=y_2) → horizontal line → (m=0).
Parallel lines: equal slopes.
Perpendicular lines: product of slopes (-1) (provided neither is vertical).
5. Equation of a Straight Line
5.1 Point‑Slope Form
Given a point ((x_1, y_1)) on the line and slope (m),
[ (y - y_1) = m (x - x_1) ]
Derivation: The change in (y) from ((x_1, y_1)) to ((x, y)) must equal (m) times the change in (x).
5.2 General Form
[ Ax + By + C = 0 ]
where (A, B, C \in \mathbb{R}) and not both (A) and (B) are zero.
Relationship to slope
If (B \neq 0), then (m = -\frac{A}{B}).
If (B=0), the line is vertical: (x = -\frac{C}{A}).
5.3 Edge Cases
| Case | Equation | Notes |
|---|---|---|
| Vertical | (x = a) | Slope undefined. |
| Horizontal | (y = b) | Slope (=0). |
| Degenerate line | (0x + 0y + C = 0) | Only possible if (C=0), giving the * |