Table of Contents

Toggle## The slope of a line

(i) Slope (m) = tan θ

(ii) Slope (m) = y_2-y_1\over x_2-x_1

**Important result**

- The slope of a line parallel to the x-axis, m=0
- The slope of a line parallel to the y-axis, m=∞

## Equation of a straight Line

(i) The general equation of a straight line is Ax + By + C = 0

(ii) The equation of a line parallel to the x-axis at a distance b from it, is given by y = b

(iii) The equation of a line parallel to the x-axis at a distance a from it, is given by x = a

(iv) Equation of the X-axis is y = 0

(v) Equation of the Y-axis is x = 0

## Different Forms of the Straight Line

(i) **Slope Intercept Form: **The equation of a line with slope *m* and making an intercept *c* on the Y-axis, is

y = mx + c

(ii) **One Point Slope Form:** The equation of a line which passes through the point (x_{1}, y_{1}) and has the slope *m* is given by

y – y_{1} = m(x-x_{1}) + c

(iii) **Two Points Form: **The equation of a line which passes through the point (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by

(y-y_{1}) = y_2-y_1\over x_2-x_1(x-x_{1})

(iv) **Intercept Form: **The equation of a line which cuts off intercept a and b respectively on the X and Y axis is given by

{x\over a}+{y\over b}=1

The general equation Ax+By+C=0 can be converted into intercept form, as

{x\over -C/A}+{y\over -C/B}=1

(v) **Normal Form: **The equation of a straight line upon which the length of the perpendicular from the origin is p and the angle made by this perpendicular to the X-axis is α, is given by

x cos α + y sin α = p

(vi) **Distance and Parametric Form**: The equation of a straight line passing through (x_{1}, y_{1}) and making an angle θ with the positive direction of X-axis, is

{x - x_1\over cos θ} = {y - y_1\over sin θ} = r

## Condition of concurrency

Condition of concurrency for three given lines a_{1}x + b_{1}y+c_{1} = 0, a_{2}x + b_{2}y+c_{2} = 0 and a_{3}x + b_{3}y+c_{3} = 0 is

a_{3}(b_{1}c_{2 }– b_{2}c_{1})+b_{3}(c_{1}a_{2 }– a_{1}c_{2})+c_{3}(a_{1}b_{2 }– a_{2}b_{1})=0

or, \begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{vmatrix}

## Distance of a point from a line

The distance of a point from a line is the length of the perpendicular drawn from the point to the line. let Ax+By+C=0 be a line, whose perpendicular distance from the point P(x_{1}, y_{1}) is d. then,

d = |Ax_1+By_1+C|\over \sqrt{A^2+B^2}

* Note:* The distance of the origin from the line

d = |C|\over \sqrt{A^2+B^2}

## Distance between two parallel Lines

The distance between two parallel lines

y = mx + c_{1}

y = mx + c_{2}

d = |c_1-c_2|\over \sqrt{1+m^2}