# Straight Line Formulae

Straight line formulae will help us to solve different types of problems on straight line in co-ordinate geometry.

1. If a straight line makes an angle α with the positive direction of the x-axis then the slope or gradient of the line i.e. m = tan α.

2. Slope of the line joining the points (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) is

m = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$ = $$\frac{\textrm{Difference of ordinates of the given point}}{\textrm {Difference of abscissa of the given point}}$$

3. Condition of collinearity of three points (x$$_{1}$$, y$$_{1}$$), (x$$_{2}$$, y$$_{2}$$) and (x$$_{3}$$, y$$_{3}$$) is x$$_{1}$$ (y$$_{2}$$  - y$$_{3}$$) + x$$_{2}$$ (y$$_{3}$$ - y$$_{1}$$) + x$$_{3}$$ (y$$_{1}$$ - y$$_{2}$$) = 0.

4. The equation of x-axis is y = 0.

5. The equation of y-axis is x = 0.

6. The equation of the line parallel to x-axis at a distance h units from x-axis is, y = h.

7. The equation of the line parallel to y-axis at a distance k units from y-axis is, x = k.

8. The equation of a straight line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.

9. The equation of a straight line in point-slope form is y - y$$_{1}$$ = m (x - x$$_{1}$$) where m is the slope of the line and (x$$_{1}$$, y$$_{1}$$) is a given point on the line.

10.The equation of a straight line in symmetrical form is

$$\frac{\mathrm{x - x_{1}}}{\textrm{cos} \mathrm{\theta}}$$ = $$\frac{\mathrm{y - y_{1}}}{\textrm{sin} \mathrm{\theta}}$$ = r

Where θ is the inclination of the line, (x$$_{1}$$, y$$_{1}$$) is a given point on the line and r is the distance between the points (x, y) and (x$$_{1}$$, y$$_{1}$$).

11. The equation of a straight line in distance form is

$$\frac{\mathrm{x - x_{1}}}{\textrm{cos} \mathrm{\theta}}$$ = $$\frac{\mathrm{y - y_{1}}}{\textrm{sin} \mathrm{\theta}}$$ = r

Where θ is the inclination of the line, (x$$_{1}$$, y$$_{1}$$) is a given point on the line and r is the distance of the point (x, y) on the line from the point (x$$_{1}$$, y$$_{1}$$).

12. The equation of a straight line in two-point form is

$$\frac{y - y_{1}}{x - x_{1}}$$ = $$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$$ or, y - y$$_{1}$$ = $$\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$$ (x - x$$_{1}$$)

Where (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) are two given points on the line.

13. The equation of a straight line in intercept form is $$\frac{x}{a}$$ + $$\frac{y}{b}$$ = 1

Where a is the x-intercept and b is the y-intercept of the line. The straight line intersects the x-axis at (a, 0) and y-axis at (0, b).

14. The equation of a straight line in normal form is x cos α + y sin α = p where p (> 0) is the perpendicular distance of the line from the origin and a (0 ≤ α ≤ 2π) is the angle that the drawn perpendicular on the line makes with the positive direction of the x-axis.

15. The equation of a straight line in general form is ax + by + c = 0 where a, b and c are real constants (a and b both are not zero).

16. To find the co-ordinates of the point of intersection of two given lines we solve the equations; the value of x is the abscissa and that of y is the ordinate of the point of intersection.

17. The equation of any straight line through the point of intersection of the lines a$$_{1}$$x + b$$_{1}$$y + c$$_{1}$$ = 0  and a$$_{2}$$x + b$$_{2}$$y + c$$_{2}$$ = 0  is

a$$_{1}$$x + b$$_{1}$$y + c$$_{1}$$ + λ (a$$_{2}$$x + b$$_{2}$$y + c$$_{2}$$) = 0, where λ(≠ 0 or ∞) is an arbitrary constant.

18. Three given straight lines are concurrent if the point of intersection of any two of them satisfies the equation of the third straight line.

19. If θ be the acute angle between the straight lines y = m$$_{1}$$x + c$$_{1}$$ and y = m$$_{2}$$x + c$$_{2}$$ then,

tan θ = |$$\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$$| or, tan θ = ± $$\frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}$$

20. If two straight lines are parallel then their slopes would be equal. Thus, the condition of parallelism for the lines y = m$$_{1}$$x+ c$$_{1}$$ and y = m$$_{2}$$x + c$$_{2}$$ is, m$$_{1}$$ = m$$_{2}$$.

21. The equation of any straight line parallel to the line ax + by + c 0 is ax + by = k where k is an arbitrary constant.

22. Two straight lines are perpendicular to each other if the product of ,their slopes = – 1. Thus, the condition of perpendicularity of the lines y = m$$_{1}$$x + c$$_{1}$$ and y = m$$_{2}$$x + c$$_{2}$$ is m$$_{1}$$ m$$_{2}$$ = - 1.

23. The equation of any straight line perpendicular to the line ax + by + c = 0 is bx - ay = k where k is an arbitrary constant.

24. The two equations a$$_{1}$$ x + b$$_{1}$$ y + c$$_{1}$$ = 0 and a$$_{2}$$ x + b$$_{2}$$y + c$$_{2}$$  = 0 represent the equation of the same straight line when $$\frac{a_{1}}{a_{2}}$$ = $$\frac{b_{1}}{b_{2}}$$ = $$\frac{c_{1}}{c_{2}}$$.

25. Let ax + by + c = 0 be a given straight line and P (x$$_{1}$$, y$$_{1}$$) and Q (x$$_{2}$$, y$$_{2}$$), two given points. The points P and Q are on the same side or opposite sides of the line ax + by + c = 0 according as (ax + by + c) and (ax$$_{1}$$ + by$$_{1}$$ + c) are of the same or opposite signs.

The origin and the point P (x$$_{1}$$, y$$_{1}$$) are on the same side or opposite sides of the straight line ax + by + c = 0 according as c and (ax$$_{1}$$ + by$$_{1}$$ + c) are of the same or opposite signs.

26. Let P (x$$_{1}$$, y$$_{1}$$) be a point not lying on the straight line ax + by + c = 0; then the length of the perpendicular drawn from P upon the line is

±$$\frac{a_{1}x + b_{1}y + c}{\sqrt{a^{2} + b^{2}}}$$ or,$$\frac{|a_{1}x + b_{1}y + c|}{\sqrt{a^{2} + b^{2}}}$$

27. The equations of the bisectors of the angles between the straight lines a$$_{1}$$x + b$$_{1}$$y + c$$_{1}$$ = 0 and a$$_{2}$$x + b$$_{2}$$y + c$$_{2}$$ = 0 are,

$$\frac{a_{1}x + b_{1}y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}}$$ = ±$$\frac{a_{2}x + b_{2}y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$$

If c$$_{1}$$ and c$$_{2}$$ are of the same signs then the bisector containing the origin is,

$$\frac{a_{1}x + b_{1}y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}}$$ = +$$\frac{a_{2}x + b_{2}y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$$.

If c$$_{1}$$ and c$$_{2}$$ are of opposite signs then the bisector containing the origin is,

$$\frac{a_{1}x + b_{1}y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}}$$ = -$$\frac{a_{2}x + b_{2}y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}$$.

The Straight Line

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Subtracting Integers | Subtraction of Integers |Fundamental Operations

Jun 13, 24 02:51 AM

Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.

2. ### Properties of Subtracting Integers | Subtraction of Integers |Examples

Jun 13, 24 02:28 AM

The properties of subtracting integers are explained here along with the examples. 1. The difference (subtraction) of any two integers is always an integer. Examples: (a) (+7) – (+4) = 7 - 4 = 3

3. ### Math Only Math | Learn Math Step-by-Step | Worksheet | Videos | Games

Jun 13, 24 12:11 AM

Presenting math-only-math to kids, students and children. Mathematical ideas have been explained in the simplest possible way. Here you will have plenty of math help and lots of fun while learning.

4. ### Addition of Integers | Adding Integers on a Number Line | Examples

Jun 12, 24 01:11 PM

We will learn addition of integers using number line. We know that counting forward means addition. When we add positive integers, we move to the right on the number line. For example to add +2 and +4…