Equation of a Straight Line in Two-Point Form

The equation of a straight line can be expressed in various forms, one of which is the two-point form. This form is particularly useful when we know the coordinates of two distinct points through which the line passes.

Two-Point Form of a Line

Given two points ( P_1(x_1, y_1) ) and ( P_2(x_2, y_2) ), the two-point form of the equation of the line passing through these points is derived from the concept of the slope of a line.

The slope ( m ) of the line is the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line:

[ m = \frac{y_2 - y_1}{x_2 - x_1} ]

Using the slope and one of the points, we can write the equation of the line as:

[ y - y_1 = m(x - x_1) ]

Substituting the value of ( m ) from the slope equation, we get the two-point form of the equation:

[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) ]

This can be rearranged to:

[ (y - y_1)(x_2 - x_1) = (y_2 - y_1)(x - x_1) ]

Which simplifies to:

[ (x_2 - x_1)y - (y_2 - y_1)x = x_2y_1 - x_1y_2 ]

Important Points and Differences

Here is a table summarizing the important points and differences related to the two-point form of a straight line:

Aspect	Description
Slope	The slope ( m ) is the measure of the steepness of the line and is calculated using the coordinates of the two points.
Two-Point Form	The equation of the line using the coordinates of two points without explicitly calculating the slope.
General Form	The two-point form can be rearranged to the general form ( Ax + By + C = 0 ), where ( A, B, ) and ( C ) are constants.
Applicability	The two-point form is only applicable when two distinct points on the line are known.
Limitations	If the two points have the same x-coordinate, the slope is undefined, and the line is vertical. The two-point form cannot be used directly in this case.

Examples

Let's go through a couple of examples to illustrate the use of the two-point form of a straight line.

Example 1

Find the equation of the line passing through the points ( P_1(2, 3) ) and ( P_2(5, 7) ).

Solution:

Using the two-point form:

[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) ]

Substitute the given points:

[ y - 3 = \frac{7 - 3}{5 - 2}(x - 2) ]

[ y - 3 = \frac{4}{3}(x - 2) ]

[ 3(y - 3) = 4(x - 2) ]

[ 3y - 9 = 4x - 8 ]

[ 4x - 3y + 1 = 0 ]

So, the equation of the line in general form is ( 4x - 3y + 1 = 0 ).

Example 2

Determine the equation of the line that passes through ( P_1(-1, 4) ) and ( P_2(3, -2) ).

Solution:

Applying the two-point form:

[ y - 4 = \frac{-2 - 4}{3 - (-1)}(x - (-1)) ]

[ y - 4 = \frac{-6}{4}(x + 1) ]

[ y - 4 = -\frac{3}{2}(x + 1) ]

[ 2(y - 4) = -3(x + 1) ]

[ 2y - 8 = -3x - 3 ]

[ 3x + 2y - 5 = 0 ]

Thus, the equation of the line is ( 3x + 2y - 5 = 0 ).

In conclusion, the two-point form of the equation of a straight line is a powerful tool for finding the equation of a line when two points on the line are known. It is straightforward to use and can be easily converted to other forms of the line equation if needed.