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
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