1. Understanding the Two-Point Form
The two-point form of a line is used when we know any two points on the line. Instead of calculating the slope separately, this formula directly uses the coordinates of both points to form the equation.
This form is useful because any two distinct points uniquely determine a straight line.
1.1. Definition
If a line passes through two points \((x_1, y_1)\) and \((x_2, y_2)\), the equation of the line can be written using the two-point form:
\( \dfrac{y - y_1}{y_2 - y_1} = \dfrac{x - x_1}{x_2 - x_1} \)
2. Understanding the Formula
The two-point form connects any point \((x, y)\) on the line to the two known points by comparing slopes.
2.1. Slope Connection
The idea behind the formula is that the slope between any point on the line and the first point \((x_1, y_1)\) must be equal to the slope between the two known points.
2.2. Geometric Meaning
This form ensures that all points \((x, y)\) satisfying the equation lie on the same straight line that passes through the two given points.
2.2.1. Special Case
If \(x_1 = x_2\), the line is vertical and the equation simplifies to:
\( x = x_1 \)
3. How to Use the Two-Point Form
To apply the two-point form, substitute the coordinates of the two points directly into the formula, and simplify if required.
3.1. Steps
- Identify the two points: \((x_1, y_1)\) and \((x_2, y_2)\).
- Write the expression:
\( \dfrac{y - y_1}{y_2 - y_1} = \dfrac{x - x_1}{x_2 - x_1} \)
- Simplify the equation to get the final linear form (optional).
4. Examples
The following examples show how to apply the two-point form in different cases.
4.1. Example 1: Basic Example
Find the equation of the line passing through \((1, 2)\) and \((3, 6)\).
\( \dfrac{y - 2}{6 - 2} = \dfrac{x - 1}{3 - 1} \)
This simplifies to:
\( \dfrac{y - 2}{4} = \dfrac{x - 1}{2} \)
4.2. Example 2: Negative Slope
Find the line passing through \((4, 5)\) and \((8, -3)\).
\( \dfrac{y - 5}{-3 - 5} = \dfrac{x - 4}{8 - 4} \)
Simplifies to:
\( \dfrac{y - 5}{-8} = \dfrac{x - 4}{4} \)
4.3. Example 3: Vertical Line
Points: \((6, 2)\) and \((6, 9)\)
Since the x-coordinates are the same:
\( x = 6 \)
4.4. Example 4: Converting to Slope-Intercept Form
Starting from Example 1:
\( \dfrac{y - 2}{4} = \dfrac{x - 1}{2} \)
Cross-multiply:
\( 2(y - 2) = 4(x - 1) \)
Expand:
\( 2y - 4 = 4x - 4 \)
So the equation becomes:
\( y = 2x \)