# 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