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

• Straight Line
• Slope of a Straight Line
• Slope of a Line through Two Given Points
• Collinearity of Three Points
• Equation of a Line Parallel to x-axis
• Equation of a Line Parallel to y-axis
• Slope-intercept Form
• Point-slope Form
• Straight line in Two-point Form
• Straight Line in Intercept Form
• Straight Line in Normal Form
• General Form into Slope-intercept Form
• General Form into Intercept Form
• General Form into Normal Form
  
• Point of Intersection of Two Lines
  
• Concurrency of Three Lines
• Angle between Two Straight Lines
• Condition of Parallelism of Lines
• Equation of a Line Parallel to a Line
• Condition of Perpendicularity of Two Lines
• Equation of a Line Perpendicular to a Line
• Identical Straight Lines
• Position of a Point Relative to a Line
• Distance of a Point from a Straight Line
• Equations of the Bisectors of the Angles between Two Straight Lines
• Bisector of the Angle which Contains the Origin
• Straight Line Formulae
• Problems on Straight Lines
• Word Problems on Straight Lines
• Problems on Slope and Intercept

11 and 12 Grade Math 

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