Straight line in Two-point Form

We will learn how to find the equation of a straight line in two-point form or the equation of the straight line through two given points.

The equation of a line passing through two points (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)) is y - y\(_{1}\) = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\)(x - x1)

Let the two given points be (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)).

We have to find the equation of the straight line joining the above two points.

Let the given points be A (x\(_{1}\), y\(_{1}\)), B (x\(_{2}\), y\(_{2}\)) and P (x, y) be any point on the straight line joining the points A and B.

Now, the slope of the line AB is \(\frac{y_{1} - y_{2}}{x_{1} - x_{2}}\)

And the slope of the line AP is \(\frac{y - y_{1}}{x - x_{1}}\)

But the three points A, B and P are collinear.

Therefore, slope of the line AP = slope of the line AB

⇒ \(\frac{y - y_{1}}{x - x_{1}}\) = \(\frac{y_{1} - y_{2}}{x_{1} - x_{2}}\)

⇒ y - y\(_{1}\) = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\) (x - x\(_{1}\))

The above equation is satisfied by the co-ordinates of any point P lying on the line AB and hence, represents the equation of the straight line AB.


Solved examples to find the equation of a straight line in two-point form:

1. Find the equation of the straight line passing through the points (2, 3) and (6, - 5).

Solution:

The equation of the straight line passing through the points (2, 3) and (6, - 5) is

 \(\frac{ y - 3}{ x + 2}\) =  \(\frac{3 + 5}{2 - 6}\),[Using the form,  \(\frac{y - y_{1}}{x - x_{1}}\) = \(\frac{y_{1} - y_{2}}{x_{1} - x_{2}}\)]

⇒ \(\frac{ y - 3}{ x + 2}\) = \(\frac{ 8}{ -4}\)

⇒ \(\frac{ y - 3}{ x + 2}\) = -2

⇒ y - 3 = -2x - 4

⇒ 2x + y + 1 = 0, which is the required equation


2. Find the equation of the straight line joining the points (- 3, 4) and (5, - 2).

Solution:

Here the given two points are (x\(_{1}\), y\(_{1}\)) = (- 3, 4) and (x\(_{2}\), y\(_{2}\)) = (5, - 2).

The equation of a line passing through two points (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)) is y - y\(_{1}\) = [\(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\)](x - x\(_{1}\)).

So the equation of the straight line in two point form is

y - y\(_{1}\) = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\) (x - x\(_{1}\))

⇒ y - 4 = \(\frac{-2 - 4}{5 - (-3)}\)[x - (-3)]

⇒ y - 4 = \(\frac{ -6}{ 8}\)(x + 3)

⇒ y - 4 = \(\frac{ -3}{ 4}\)(x + 3)

⇒ 4(y - 4) = -3(x + 3)

⇒ 4y - 16 = -3x - 9

⇒ 3x + 4y - 7 = 0, which is the required equation.






11 and 12 Grade Math

From Straight line in Two-point Form 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.