Coordinate Geometry – Class 9 Mathematics

Coordinate Geometry studies points, lines, and figures using coordinates on the Cartesian plane.

1. Cartesian Plane

• A plane is divided into four quadrants by:

  • x-axis (horizontal)

  • y-axis (vertical)

• Any point P is represented as $P(x, y)$, where:

  • $x$ = abscissa (distance from y-axis)

  • $y$ = ordinate (distance from x-axis)

Quadrants:

2. Distance Formula

Distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^{\mathrm{2<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"></span>}}}$$

3. Midpoint Formula

Midpoint $M$ of line segment joining $P(x_1, y_1)$ and $Q(x_2, y_2)$:

$$M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

4. Section Formula

Point dividing the line joining $P(x_1, y_1)$ and $Q(x_2, y_2)$ in the ratio $m:n$:

• Internally:

$$(x, y) = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right)$$

• Externally:

$$(x, y) = \left(\frac{m x_2 – n x_1}{m – n}, \frac{m y_2 – n y_1}{m – n}\right)$$

5. Slope of a Line

Slope $m$ of line joining $P(x_1, y_1)$ and $Q(x_2, y_2)$:

$$m=\frac{y_2-y_1}{x_2-x_1},x_2\mathrm{\ne }{x}_1$$

• Horizontal line: $m = 0$
  

• Vertical line: $m = \text{undefined}$
  

6. Equation of a Line

1. Slope-intercept form:

$$y=mx+c$$

• $m$ = slope

• $c$= y-intercept

2. Point-slope form:

$$y – y_1 = m(x – x_1)$$

• Passes through $(x_1, y_1)$

3. Two-point form:

$$y – y_1 = \frac{y_2 – y_1}{x_2 – x_1}(x – x_1)$$

4. Intercept form:

If line cuts x-axis at $a$ and y-axis at $b$:

$$\frac{x}{a} + \frac{y}{b} = 1$$

7. Distance of a Point from a Line

For line $Ax + By + C = 0$ and point $P(x_1, y_1)$:

$$d=\frac{\mathrm{\mid }A{x}_1+B{y}_1+C\mathrm{\mid }}{\sqrt{A^2+B^{\mathrm{2<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"></span>}}}}$$

8. Collinearity of Points

Three points $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)$ are collinear if:

$$\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_2}{x_3-x_{\mathrm{2<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"></span><span\; class="vlist-s"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"></span>}}}$$

Or, using area method:

$$\text{Area of triangle ABC}=\frac{1}{2}\mid \begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\mid =0$$

9. Area of Triangle (Using Coordinates)

Area of triangle with vertices $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)$:

$$\text{Area}=\frac{1}{2}\mid x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\mid$$

10. Important Notes

• Distance, midpoint, and slope formulas are frequently used together.

• Always check the order of points when using formulas.

• Coordinate Geometry is a bridge between algebra and geometry.