Straight line formulae will help us to solve
different types of problems on straight line in coordinate geometry.
1. If a straight line makes an angle α with the positive direction of the xaxis 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 xaxis is y = 0.
5. The equation of yaxis is x = 0.
6. The equation of the line parallel to xaxis at a distance h units from xaxis is, y = h.
7. The equation of the line parallel to yaxis at a distance k units from yaxis is, x = k.
8. The equation of a straight line in slopeintercept form is y = mx + b, where m is the slope of the line and b is the yintercept.
9. The equation of a straight line in pointslope 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 twopoint 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 xintercept and b is the yintercept of the line. The straight line intersects the xaxis at (a, 0) and yaxis 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 xaxis.
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 coordinates 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,
\(\frac{a_{1}x + b_{1}y + c_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2}}}\) = \(\frac{a_{2}x + b_{2}y + c_{2}}{\sqrt{a_{2}^{2} + b_{2}^{2}}}\).
11 and 12 Grade Math
From Straight Line Formulae to HOME PAGE
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.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.