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 x-axis is x = 0

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

12. The equation of the line parallel to the y-axis and at a distance b isx = 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 bas 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 parallel, 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}} }}

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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