# Formulas and Results of Straight Lines

Consider two points $P\left( {{x_1},{y_1}} \right)$ and $Q\left( {{x_2},{y_2}} \right)$ then

01. The distance formula

02. Midpoint formula

03. The point $R\left( {x,y} \right)$ dividing $\overline {PQ}$ in the ratio $\frac{{{k_1}}}{{{k_2}}}$ is

04. Slope of $\overline {PQ}$ is

05. Slope of x-axis = zero

06. Slope of line parallel to x-axis = zero

07. Slope of y-axis is not defined i.e. $\infty$

08. Slope of line parallel to y-axis is not defined i.e. $\infty$

09. Equation of x-axis is $y = 0$

10.  Equation of x-axis is $x = 0$

11. Equation of the line parallel to x-axis and at a distance $a$ is$y = a$.

12. Equation of the line parallel to y-axis and at a distance $b$ is$x = b$.

13. Equation of the line with slope $m$ and y-intercept $c$ is $y = mx + c$, which is called slope – intercept form.

14. Equation of the line passing through $\left( {{x_1},{y_1}} \right)$ and having the slope $m$ is $y - {y_1} = m\left( {x - {x_1}} \right)$, which is called slope – point form.

15. Equation of the line passing through two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is

.

16. Equation of the line having $a$ and $b$as x – intercept and y – intercept is $\frac{x}{a} + \frac{y}{b} = 1$ and is called the equation of the line in intercept form.

17. Normal form of the straight line is $x\cos \alpha + y\sin \alpha = p$, where $p$ is the length of the perpendicular from $O\left( {0,0} \right)$ to the line, and $\alpha$ is the inclination of the perpendicular.

18. General form of the equation of a straight line is $ax + by + c = 0$ Consider two lines ${l_1}$ and ${l_2}$ having the slopes ${m_1}$ and ${m_2}$ respectively.

19. If two lines ${l_1}$ and ${l_2}$ are parallel, then ${m_1} = {m_2}$.

20. If two lines ${l_1}$ and ${l_2}$ are parallel, then ${m_1} \times {m_2} = - 1$.

21. Angle $\theta$ from ${l_1}$ to ${l_2}$ is

22. Distance of point $\left( {{x_1},{y_1}} \right)$ from the line $ax + by + c = 0$ is

.

23. If $ax + by + c = 0$ with $b > 0$, is the equation of the line $l$, the $P\left( {{x_1},{y_1}} \right)$ lies:
(1) above the line $l$ if $a{x_1} + b{y_1} + c > 0$
(2) below the line $l$ if $a{x_1} + b{y_1} + c < 0$
(3) on the line $l$ if  $a{x_1} + b{y_1} + c = 0$

24. Three lines ${a_1}x + {b_1}y + {c_1} = 0$, ${a_2}x + {b_2}y + {c_2} = 0$, ${a_3}x + {b_3}y + {c_3} = 0$ are concurrent if