# 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

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