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. 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. Slope of \overline {PQ} is

m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

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 isy = a.

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

\frac{{y - {y_1}}}{{{y_2}  - {y_1}}} = \frac{{x - {x_1}}}{{{x_2} - {x_1}}}

.

16. Equation of the line having a and bas 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

\tan \theta = \frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}

22. 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

Comments

comments

Posted in: