U25 Coordinate Geometry (Advanced) Essential Formulas

This section covers advanced topics in coordinate geometry, including the distance formula, section formula, area of polygons, equations of straight lines, and the relationship between lines.

1 Distance and Section Formulas

Distance Between Two Points

The distance $d$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Section Formula (Internal Division)

If a point $P$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m : n$, then the coordinates of $P$ are:

$$ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$

2 Area of Polygons

Area of a Triangle

The area $\Delta$ of a triangle with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ is given by the determinant formula. The area is always taken as positive.

$$ \Delta = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - x_2y_1 - x_3y_2 - x_1y_3 \right| $$

Collinearity of Three Points

Three points $A$, $B$, and $C$ are collinear if and only if the area of the triangle formed by them is zero.

$$ \text{Area} = 0 \quad \Leftrightarrow \quad x_1y_2 + x_2y_3 + x_3y_1 - x_2y_1 - x_3y_2 - x_1y_3 = 0 $$

3 Equations of Straight Lines

General Form and Slope-Intercept Form

The general form is $Ax + By + C = 0$, where $A$, $B$, $C$ are constants and $A$ and $B$ are not both zero. The slope-intercept form is $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

$$ Ax + By + C = 0 \quad \text{or} \quad y = mx + c $$

Point-Slope Form

The equation of a line with slope $m$ passing through a point $(x_1, y_1)$ is:

$$ y - y_1 = m(x - x_1) $$

Two-Point Form

The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} $$

4 Relationship Between Lines

Parallel and Perpendicular Lines

For two lines with slopes $m_1$ and $m_2$:

They are parallel if and only if $m_1 = m_2$.
They are perpendicular if and only if $m_1 \times m_2 = -1$.

$$ \text{Parallel: } m_1 = m_2 \qquad \text{Perpendicular: } m_1 m_2 = -1 $$

Angle Between Two Lines

If $\theta$ is the acute angle between two lines with slopes $m_1$ and $m_2$, then:

$$ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| $$

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