# 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 $\left| {PQ} \right| = \sqrt {{{\left( {{x_2} – {x_1}} \right)}^2} + {{\left( {{y_2} – {y_1}} \right)}^2}}$

02. The midpoint formula $\overline {PQ} = \left( {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)$

03. The point $R\left( {x,y} \right)$ dividing $\overline {PQ}$ in the ratio $\frac{{{k_1}}}{{{k_2}}}$ is $x = \frac{{{k_1}{x_2} + {k_2}{x_1}}}{{{k_1} + {k_2}}},\,\,\,y = \frac{{{k_1}{y_2} + {k_2}{y_1}}}{{{k_1} + {k_2}}}$

04. The slope of $\overline {PQ}$ is $m = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}$

05. The slope of the x-axis = zero

06. The slope of a line parallel to the x-axis = zero

07. The slope of the y-axis is not defined, i.e. $\infty$

08. The slope of a line parallel to the y-axis is not defined, i.e. $\infty$

09. The equation of the x-axis is $y = 0$

10.  The equation of the y-axis is $x = 0$

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

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

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

14. The 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 the slope – point form.

15. The equation of the line passing through two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\frac{{y – {y_1}}}{{{y_2} – {y_1}}} = \frac{{x – {x_1}}}{{{x_2} – {x_1}}}$.

16. The equation of the line having $a$ and $b$as the 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. The 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. The 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 perpendicular, then ${m_1} \times {m_2} = – 1$.

21. The angle $\theta$ from ${l_1}$ to ${l_2}$ is $\tan \theta = \frac{{{m_2} – {m_1}}}{{1 + {m_1}{m_2}}}$

22. The distance of point $\left( {{x_1},{y_1}} \right)$ from the line $ax + by + c = 0$ is $\frac{{\left| {a{x_1} + b{y_1} + c} \right|}}{{\sqrt {{a^2} + {b^2}} }}$.

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

$\left| \begin{array}{ccc} {a_1} & {b_1} & {c_1} \\ {a_2} & {b_2} & {c_2} \\ {a_3} & {b_3} & {c_3} \end{array} \right| = 0$