Coordinate geometry formulae:
- The distance between two points with coordinates $$(x_1, y_1), (x_2, y_2)$$ is given by $$ d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
- Point slope form of the equation of a line is $$y-y_1=m(x-x_1)$$
- Two point form of the equation of a line is $$y-y_1=(y_2-y_1)\left(\frac{x-x_1}{x_2-x_1}\right)$$
- Mid point between two points $$A\left(x_1,y_1\right)$$ and $$B\left(x_2,y_2\right)$$ is $$\left(\frac{\left(x_1+x_2\right)}{2},\frac{\left(y_1+y_2\right)}{2}\right)$$
- Coordinates of a point P that divides the line joining $$A\left(x_1,y_1\right)$$ and $$B\left(x_2,y_2\right)$$ internally in the ratio l:m : $$\left(\frac{\left(lx_2+mx_1\right)}{l+m},\frac{\left(ly_2+my_1\right)}{l+m}\right)$$.
- Coordinates of a point P that divides the line joining $$A\left(x_1,y_1\right)$$ and $$B\left(x_2,y_2\right)$$ externally in the ratio l:m : $$\left(\frac{\left(lx_2-mx_1\right)}{l-m},\frac{\left(ly_2-my_1\right)}{l-m}\right)$$.
- A line can be defined as $$y=mx+c$$ where m is the slope of the line and c is the y-intercept.
- Slope $$m=\frac{\left(y_2-y_1\right)}{x_2-x_1}$$. Here, if $$x_2=x_1$$, then the two lines are perpendicular to each other.
- When two lines are parallel, their slopes are equal i.e $$m_1=m_2$$
- When two lines are perpendicular, product of their slopes = -1 i.e $$m_1*m_2=-1$$
- If a and b are the x and y intercept of a line then $$\frac{x}{a}+\frac{y}{b}=1$$
- If two intersecting lines have slopes $$m_1$$ and $$m_2$$, then the angle between the two lines will be $$\tan\theta\ =\frac{\left(m_1-m_2\right)}{1+m_1m_2}$$.
- The length of perpendicular from a point $$\left(X_1,Y_1\right)$$ on the line AX+BY+C=0 is $$\frac{\left(AX_1+BY_1+C\right)}{\sqrt{\ A^2+B^2}}$$.
- The distance between two parallel lines Ax+By+C1 = 0 and Ax+By+C2= 0 is $$\left|\frac{C_1-C_2}{\sqrt{\ A^2+B^2}}\right|$$
- Image of the point (m,n) in the line ax + by + c = 0 is given by $$\dfrac{\left(x-m\right)}{a}=\dfrac{\left(y-n\right)}{b}=-\dfrac{2\left(am+bn+c\right)}{a^2+b^2}$$
