Coordinate Geometry

The Cartesian Plane — Quick Recap

The Cartesian plane has two perpendicular axes: the horizontal x-axis and the vertical y-axis. Every point is identified by an ordered pair $(x, y)$. The point where the axes meet is the origin $(0, 0)$. The four quadrants are: Q1 (+,+), Q2 (-,+), Q3 (-,-), Q4 (+,-).

Distance Formula

The distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is: $PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

This is derived from the Pythagoras theorem — the horizontal distance $(x_2-x_1)$ and vertical distance $(y_2-y_1)$ form the two legs of a right triangle, and PQ is the hypotenuse.

Special case: Distance from origin: $OP = \sqrt{x^2 + y^2}$.

Section Formula

If 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:

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

Midpoint (m:n = 1:1): $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

To find the ratio in which a point divides a segment: let the ratio be $k:1$, substitute, and solve for $k$.