Coordinate Geometry

Gradient

$m=\frac{y_1-y_2}{x_1-x_2}$

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

$y=mx+c$

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

Parallel Lines

$m_1=m_2$

Perpendicular Lines

$m_1 \times m_2=-1$

$$A=\frac{1}{2}\left| \begin{array}{ccccc} x_1 & x_2 & x_3 & x_4 & x_1\\ y_1& y_2 & y_3 & y_4 & y_1\end{array} \right|$$

$=\frac{1}{2}|(x_1y_2+x_2y_3+x_3y_4+x_4y_1)-(x_2y_1+x_3y_2+x_4y_3+x_1y_4)|$

Note: Coordinates should be in anti-clockwise direction

Standard Form

$(x-a)^2+(y-b)^2=r^2$

$(a,b)$: centre of circle

$r$: radius

General Form

$x^2+y^2+2gx+2fy+c=0$

$(-g,-f)$: centre of circle

$\sqrt{g^2+f^2-c}$: radius