SHORT NOTES: Straight Lines

Equation of a straight line in various forms:
A line is defined as locus of points satisfying the condition "ax+by+c=0" where a,b,c are constants.
(i)The slope intercept form:
Slope : $m=\tan\theta=\frac{y}{x}$

Line with slope 'm' and intercept 'c' on y axis is ${\color{DarkRed} y=mx+c}$

(ii)The point slope form:

Line passing through point $\left (x_1,y_1 \right )$ and has slope 'm' is $y-y_1=m\left ( x-x_1 \right )$

(iii)The two point form:

Line passing through two points $\left ( x_1,y_1 \right )$ and $\left ( x_2,y_2 \right )$ is $y-y_1=\left ( \frac{y_2-y_1}{x_2-x_1} \right )\left ( x-x_1 \right )$

(iv)The intercept form :

$\frac{x}{a}+\frac{y}{b}=1$

Angle between two lines:

The acute angle between the lines is $\tan \theta= \left |\frac{m_2-m_1}{1+m_1m_2} \right |$ where $m_1,m_2$ are slopes.

Condition for parallelism of lines: $m_1=m_2$
Condition for perpendicular of two lines: $m_1m_2=-1$

Distance of a point from a line:
The length of perpendicular from a point $\left ( x_1,y_1 \right )$ to a line ax+by+c=0 is

$\left | \frac{ax_1+by_1+c}{\sqrt{{a^2+b^2}}} \right |$

Lines through the point of intersection of two given lines:

lines passing through intersection of two lines $a_1x+b_1y+c_1 \ a_2x+b_2y+c_2=0$ is

$\left (a_1x+b_1y+c_1 \right )+\lambda \left (a_2x+b_2y+c_2 \right )=0$

Equation of the bisector of the angle between two lines:

$\frac{a_1x+b_1y+c_1}{\sqrt{{a_1^2+b_1^2}}}=\pm \frac{a_2x+b_2y+c_2}{\sqrt{{a_2^2+b_2^2}}}$

Rewrite the equation making constants $c_1.c_2$ positive
for + sign it is the bisector of angle that contains origin.
If $a_1a_2+b_1b_2> 0$ ,obtuse angle contains origin and vice versa.

Concurrency of lines: The lines

$a_1x+b_1y+c_1=0\\a_2x+b_2y+c_2=0\\a_3x+b_3y+c_3=0$

are concurrent if

$\lambda _1\left (a_1x+b_1y+c_1 \right )+\left \lambda _2(a_2x+b_2y+c_2 \right )+\lambda _3\left (a_3x+b_3y+c_3 \right )=0$

$\begin{vmatrix} a_1 & b_1& c_1\\ a_2 & b_2 & c_2\\ a_3& b_3 & c_3 \end{vmatrix}=0$

Some standard points of a triangle:

(i)Centroid: Point of intersection of medians

$\left ( \frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3} \right )$

(ii)Incentre:Point of intersection of the internal bisectors of the angles of triangle

$\left ( \frac{ax_1+bx_2+cx_3}{a+b+c},\frac{ay_1+by_2+cy_3}{a+b+c} \right )$

(iii)Circumcentre:Point of intersection of perpendicular bisector of its sides

$\left ( \frac{x_1\sin 2A+x_2\sin 2B+x_3\sin 2C}{\sin 2A+\sin 2B+\sin 2C},\frac{y_1\sin 2A+y_2\sin 2B+y_3\sin 2C}{\sin 2A+\sin 2B+\sin 2C} \right )$

(iv)Orthocentre:Point of intersection of its altitudes

$\left ( \frac{x_1\tan A+x_2\tan B+x_3\tan C}{\tan A+\tan B+\tan C},\frac{y_1\tan A+y_2\tan B+y_3\tan C}{\tan A+\tan B+\tan C} \right )$

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Straight Lines

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