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Attached pdf of this Article as SPOILER at the top! Happy learning!
Hi All,
I have recently uploaded a video on YouTube to discuss Equation of a Line in Detail:
Following is covered in the video
¤ Equation of a Line: Two Point Form ¤ Equation of a Line: Point and Slope Form ¤ Equation of a Line: Intercept Form ¤ Generic Equation of a line (Point and Intercept Form) ¤ Equation of Horizontal and Vertical Lines
Equation of a Line: Two Point Form
Let's say we have a line passing through two point A(\(x_1\),\(y_1\)) and B(\(x_2\),\(y_2\)). Let's take a point C (x,y) on the line and between A and B as shown below.
Attachment:
EL img-1.jpg [ 20.58 KiB | Viewed 605 times ]
Slope of the Part CA of the line = Slope of the part BA of the line = Slope of the line
Equation of a line in Point and Slope Form as y - \(y_1\) = m * (x - \(x_1\))
Equation of a Line: Intercept Form
Let's say we have a line which intercepts X-Axis at point A(a,0) and Y-Axis at point B(0,b), as shown below.
Attachment:
EL img-2.jpg [ 20.17 KiB | Viewed 623 times ]
Using, \(\frac{y - y_1}{x - x_1}\) = \(\frac{y_2 - y_1}{x_2 - x_1}\) we get
\(\frac{y - 0}{x - a}\) = \(\frac{b - 0}{0 - a}\) => ay = -bx + ab => bx + ay = ab
Dividing both the sides by ab we get
Equation of a line in Intercept Form as \(\frac{x}{a}\) + \(\frac{y}{b}\) = 1
Generic Equation of a line (Point and Intercept Form)
Let's say we have a line which intercepts Y-Axis at point B(0,b) and has a slope m, as shown below.
Attachment:
EL - generic equation.jpg [ 20.28 KiB | Viewed 625 times ]
Using, y - \(y_1\) = m * (x - \(x_1\)) and substituting the value of Point B we get
y - b = m * (x - 0)
Generic Equation of a line (Point and Intercept Form) as y = mx + b
where m is the slope of the line and B is the y intercept.
Equation of horizontal and vertical lines
Let's say we have a line parallel to X-Axis and intersecting Y-Axis at point B(0,b) and a line which is parallel to Y-Axis and intercepting the X-Axis at point A(a,0) as shown below
Attachment:
EL - Horizontal and parallel lines.jpg [ 21.02 KiB | Viewed 627 times ]
Equation of Horizontal Line
Now, all the points on this line will be at the same distance b from X-Axis and will have the y-coordinate as b
=> Equation of Horizontal line will be y = b [constant]
Equation of Vertical Line
Now, all the points on this line will be at the same distance a from Y-Axis and will have the x-coordinate as a
=> Equation of Vertical line will be x = a [constant]