Two Point Form
Two point form can be used to express the equation of a line in coordinate plane. The equation of a line can be found through various methods depending on the available information. The two-point form is one of the methods. This is used to find the equation of a line when two points lying on the line are given. Some other important forms to represent the line equation are slope intercept form, intercept form, point slope form, etc. Let us understand the two point form using formula and examples in the following sections.
| 1. | What is Two-Point Form? |
| 2. | Formula For Two Point Form |
| 3. | Derivation of Two Point Form Formula |
| 4. | Finding Equation of Line Using Two Point Form |
| 5. | FAQs on Two-Point Form |
What is Two-Point Form?
Two point form is one of the important forms used to represent a straight line algebraically. The equation of a line represents each and every point on the line, i.e., it is satisfied by each point on the line. The two-point form of a line is used for finding the equation of a line given two points (x\(_1\), y\(_1\)) and x\(_2\), y\(_2\)) on it.
Equation of a Line in Two-Point Form
The two-point form of a line passing through these two points is:
\(y-y_1= \frac{y_2-y_1}{x_2-x_1}(x-x_1)\) OR \(y-y_2= \dfrac{y_2-y_1}{x_2-x_1}(x-x_2)\)
Here, (x, y) represents any random point on the line and we keep ‘x’ and ‘y’ as variables.
Formula For Two Point Form
The two point formula is used to represent a line algebraically using the coordinates of two points that lie on that line. The formula for two point form can be given as,
Two Point Form: Formula
\(y-y_1= \frac{y_2-y_1}{x_2-x_1}(x-x_1)\)
OR
\(y-y_2= \frac{y_2-y_1}{x_2-x_1}(x-x_2)\)
where,
- (x, y) is an arbitrary point on the line.
- (x\(_1\), y\(_1\)) and (x\(_2\), y\(_2\)) are coordinates of points lying on the line.
Derivation of Two Point Form Formula
We can derive the two point form equation for any line given the two points lying on that line. Let us consider two fixed points A(x\(_1\), y\(_1\)) and B(x\(_2\), y\(_2\)) on the line in a coordinate plane. Let us assume that C(x, y) is any random point on the line.
Since A, B, and C lie on the same line:
Slope of \(\overleftrightarrow{AC}\) = Slope of \(\overleftrightarrow{AB}\)
Using the slope formula,
\(\frac{y-y_1}{x-x_1}= \frac{y_2-y_1}{x_2-x_1}\)
Multiplying both sides by (x – x\(_1\)),
\(y – y_1\) = \(\frac{y_2-y_1}{x_2-x_1}(x-x_1)\)
We derived the two-point form. It is used to find the equation of a line that passes through two points.
We can derive the another formula of two-point form, which is, \(y-y_2= \dfrac{y_2-y_1}{x_2-x_1}(x-x_2)\) in a similar manner.
Finding Equation of Line Using Two Point Form
As we discussed above, the equation of a line can be found using two points that lie on that line. We can follow the below-given steps while applying the two point form to find the straight-line equation.
Step 1: Note down the coordinates of the two points lying on the line as (x\(_1\), y\(_1\)) and (x\(_2\), y\(_2\)).
Step 2: Apply the two point formula given as, \(y – y_1\) = \(\frac{y_2-y_1}{x_2-x_1}(x-x_1)\).
Step 3: Simplify the obtained equation to the form, y = mx + b to represent the line.
Important Notes on Two Point Form:
- The two-point form of a line can also be written as:
\(\frac{y-y_{1}}{x-x_{1}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) OR \(\frac{y-y_{2}}{x-x_{2}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) - The equation of a vertical line passing through (a, b) is of the form (x = a).
This is an exceptional case where the two-point form cannot be used.
Thinking out of the box:
Which of the following graphs represents the equation y – 2 = [(1 – 2)/(-2 – 1)](x – 1)?
Topics Related to Two Point Form:
Examples on Two Point Form
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Example 1: Find the equation of a straight line passing through the points A (1, 2) and B (-1, 3)?
Solution:
The following figure shows the line passing through the given points:
The given two points are: A(1, 2) = (x\(_1\), y\(_1\)); B(-1, 3) = (x\(_2\), y\(_2\))
Since we know two points on the line, we use the two-point form to find its equation.
\(y – y_1\) = \(\frac{y_2-y_1}{x_2-x_1}(x – x_1)\)
\(y\) – 2 = \(\frac{3-2}{-1-1}(x-1)\)
\(y\) – 2 = \(\frac{1}{-2}(x-1)\)
Multiplying both sides by -2,
-2(y – 2) = x – 1
-2y + 4 = x – 1
x + 2y – 5 = 0
Therefore, the equation of the line is, ∴ x + 2y – 5 = 0
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Example 2: Find the y-intercept of the line passing through the points A (3, -2) and B (-1, 3)?
Solution:
The following figure shows the line passing through the given points:
The given two points are: A(3, -2) = (x\(_1\), y\(_1\)); B(-1, 3) = (x\(_2\), y\(_2\))
Since we know two points on the line, we use the two-point form to find its equation.
\(y – y_1\) = [(y\(_2\) – y\(_1\))/(x\(_2\) – x\(_1\))](x – x\(_1\))
y + 2 = [(3 + 2)/(-1 – 3)](x – 3)
y + 2 = (5/(-4))(x – 3)
Multiplying both sides by -4,
-4(y + 2) = 5(x – 3)
-4y – 8 = 5x – 15
-4y = 5x – 7
y = (-5/4)x + (7/4)
The final equation is in the slope-intercept form, y = mx + b.
Comparing the last two equations, we get the y-intercept to be b = (7/4)
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Example 3: Find the equation of a straight line whose x-intercept is ‘a’ and y-intercept is ‘b’.
Solution:
The given line passes through the points: A(a, 0) = (x\(_1\), y\(_1\)); B(0, b) = (x\(_2\), y\(_2\))
Since we know two points on the line, we use the two-point form to find its equation.
y – y\(_1\) = [(y\(_2\) – y\(_1\))/(x\(_2\) – x\(_1\))](x – x\(_1\))
y – 0 = [(b – 0)/(0 – a)](x – a)
y = (b/-a)(x – a)
Multiplying both sides by (-a),
-ay = b(x – a)
-ay = bx – ab
bx + ay = ab
Dividing both sides by ab,
(x/a) + (y/a) = 1
Thus, the equation of the given line is:
∴ (x/a) + (y/a) = 1
Note: This is also called the intercept-form of a line.
Practice Questions on Two Point Form
FAQs on Two-Point Form
What is the Two-Point Form of a Line?
The two-point form of a line is used for finding the equation of a line given two points \((x_1,y_1)\) and \((x_2,y_2)\) on it. The two point-form of a line is:\(y-y_1= \dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\) OR \(y-y_2= \dfrac{y_2-y_1}{x_2-x_1}(x-x_2)\).
How do you Write an Equation Using Two Point Form With Two Given Points?
We substitute the points in the two-point form to find the equation. Given two points, \((x_1,y_1)\) and \((x_2,y_2)\), we can apply the two point form, \(y-y_1= \dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\) OR \(y-y_2= \dfrac{y_2-y_1}{x_2-x_1}(x-x_2)\).
How do you Find the Slope Intercept Form From Two Point Form?
The two point-form of a line is given as, \(y-y_1= \dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\) OR \(y-y_2= \dfrac{y_2-y_1}{x_2-x_1}(x-x_2)\). Substitute the value of \((x_1,y_1)\) and \((x_2,y_2)\) and rearrange to obtain the slope intercept form, y = mx + b.
What is the Example of a Two Point Form?
The equation of a line with slope, m = 1, that passes through a point \((x_1,y_1)\) = (-2, 3) using the point-slope form is:
\(y – y_1 = m(x – x_1)\)
y – 3 = 1(x + 2)
y = x + 5
How do you Find the Y Intercept With Two Points and Slope Using Two Point Form?
We can follow the steps given below to find y-intercept using two point form,
- First, find the equation of the line using the two-point form and solve it for y.
- Compare it with y = mx + b.
- Here, b is the y-intercept.
How do you Find the Slope of a Line With Two Given Points Using Two Point Form?
To find the slope of a line with two given points and two point form,
- First find the equation of line using the two-point form and solve it for y.
- Compare it with y = mx + b.
- Here, m is the slope of the line.
What is Two Point Form Formula?
The two point form formula of a line is given as, \(y-y_1= \dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\) OR \(y-y_2= \dfrac{y_2-y_1}{x_2-x_1}(x-x_2)\), where \((x_1,y_1)\) and \((x_2,y_2)\) are coordinates of two points lying on that line.
How to Derive the Two Point Form of a Straight Line?
To derive the two point form of a line, we assume any two given points lying on the line as \((x_1,y_1)\) and \((x_2,y_2)\). Using the slope formula,
\(\frac{y-y_1}{x-x_1}= \frac{y_2-y_1}{x_2-x_1}\)
⇒\(y – y_1\) = \(\frac{y_2-y_1}{x_2-x_1}(x-x_1)\)
Hence, two point form is derived for given line.
What is the Normal Form and Two Point Form of a Line?
The normal form of a line is \(x \cos \alpha+y \sin \alpha=p\). Here, \( \alpha\) is the angle made by the line with the positive direction of the x-axis and \(p\) is the perpendicular distance of the line from the origin. Also, the two point form for two points \((x_1,y_1)\) and \((x_2,y_2)\) is given as, \(y-y_1= \dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\) OR \(y-y_2= \dfrac{y_2-y_1}{x_2-x_1}(x-x_2)\).
How do you Find the Equation of a Line With Only One Point Using Two Point Form?
We can’t find the equation of a line just with one point. We need two points lying on the line to apply two point form. To find the equation of a line one of the following information should be available.
- Two points on a line.
- One point and the slope of a line.
- One point and one intercept of a line.
- Two intercepts of a line.
How do you Determine if a Point Lies On a Line?
Every point on a line satisfies its line equation. For example, to see whether (1, 2) lies on a line \(y = 2x\), we substitute \(x=1\) and \(y=2\) in the given equation. Then we get: 2 = 2(1) or, 2 = 2. The equation is satisfied and hence the point (1, 2) lies on the line y = 2x.
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