What is the External Division Formula and How to Use It with Solved Examples
The line segment is divided into two pieces by a point, which may or may not be equal. We are able to determine the ratio by which the provided line segment is divided if we are aware of the point's coordinates.
What is the Section Formula?
The coordinates of a point that separates a line segment externally or internally in a specific ratio can be found using the section formula. It's a helpful tool for figuring out the location of the point where a line segment is divided into a specific number of segments.
This section formula can be used to find the midpoint of a line segment as well as to derive the midpoint formula.
External Division Formula:
When the line segment is divided externally by the point in the specified ratio, the external section formula is applied. The coordinates of the point on the line segment connecting the two points and falling beyond the two points, in the specified ratio, are found using this formula. Look at the following illustration to see how a point P(x, y) externally divides the line segment AB in a ratio such that \[{\rm{AP:PB=m:n}}\].The coordinates of the point P are now stated as follows in terms of the ratio supplied and for the given coordinates of the points A\[\left( {{x_1},{y_1}} \right)\] , B \[\left( {{x_{2,}}{y_2}} \right)\]:
Section of a Line Dividing it in Ratio M : N
Section Formula
where,
The coordinates for point P are (x,y)
Point A's coordinates are \[\left( {{x_1},{y_1}} \right)\].
The coordinates of point B is \[\left( {{x_{2,}}{y_2}} \right)\].
The ratio values at which P divides the line externally are m and n.
Difference between Internal and External Division Formula:
This formula can be used when a point internally divides a line segment in the ratio m: n at point C and that point lies between the coordinates of the line segment. Internal Division is another name for it.
Internal Section Formula
External Division formula
We can use this formula when the point that splits the line segment is divided externally in the ratio m: n sits outside the line segment, which means that when we extend the line, the point corresponds with the line. It also goes by the name External Division.
Conclusion :
We can use this formula when the point that splits the line segment is divided externally in the ratio m: n sits outside the line segment, which means that when we extend the line, the point corresponds with the line. It also goes by the name External Division.
Solved Examples:
1. Find the ratio in which a point P lying on the y-axis divides the line joining points (6, -6) and (-2, -4).
Let , ratio be \[m:n\] .
Since the point P lies on the y-axis, the y-axis will be 0 and let us assume the x-axis to be 'a'.
So, according to the formula let us take\[\left( {6, - 6} \right) = \left({{x_1},{y_1}} \right)\]
\[\left[ { - 2, - 4} \right] = \left[ {{x_2},{y_2}} \right]\]
and \[\left[ {0,a} \right] = \left[ {x,y} \right]\]
\[x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}\]
\[\begin{array}{l}0 = - 2m + 6n\\2m = 6n\\m/n = 3/1\\m:n = 3:1\end{array}\]
2. What does the formula for internal and external sections mean?
The section formula aids in pinpointing a point's coordinates, making it easier to divide the line connecting two locations into ratios. Either internally or externally, something occurs. Following is the formula:
\[x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}\]
3. Using the section formula, determine the coordinates of the point C that externally divides the points A (-3, -1), and B (-1, 0) in a ratio of 2:3.
Given,
\[\left[ {{x_1},{y_1}} \right] = \left[ { - 3, - 1} \right]\]
\[\left( {{x_2},{y_2}} \right) = \left( { - 1,0} \right)\]
\[m:n = 2:3\]
the location of the point C's coordinates (x, y)
Using the external division formula for sections,
\[\begin{array}{l}C\left[ {x,y} \right] = \left[ {\dfrac{{\left[ {2\left[ { - 1} \right] - 3\left[ { - 3} \right]} \right]}}{{2 - 3}},\dfrac{{\left[ {2\left[ 0 \right] - 3\left[ { - 1} \right]} \right]}}{{2 - 3}}} \right]\\{\text{After solving we will get:}}\\C\left( {x,y} \right) = \left( { - 7, - 3} \right)\end{array}\]
Thus, the coordinate of point C is (-7, -3)
FAQs on External Division Formula in Coordinate Geometry
1. What is the external division formula in coordinate geometry?
The external division formula gives the coordinates of a point that divides a line segment externally in a given ratio. If a point P divides the line joining A(x₁, y₁) and B(x₂, y₂) externally in the ratio m:n, then:
P = ( (mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n) )
This formula is used in coordinate geometry to find a point lying outside the line segment AB in a specific ratio.
2. How do you find the coordinates of a point that divides a line externally in the ratio m:n?
To find the coordinates of a point dividing a line externally in the ratio m:n, use the external division formula.
- Let A(x₁, y₁) and B(x₂, y₂) be the endpoints.
- Use: P = ( (mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n) )
- Substitute the values of x₁, y₁, x₂, y₂, m, and n.
3. What is the difference between internal and external division formula?
The key difference is that internal division uses (m + n) in the denominator, while external division uses (m − n).
- Internal division: P = ( (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n) )
- External division: P = ( (mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n) )
4. Can you give an example of the external division formula?
Yes, here is a simple example using the external division formula.
- Let A(1, 2) and B(5, 6).
- Suppose P divides AB externally in the ratio 2:1.
- P = ( (2×5 − 1×1)/(2 − 1), (2×6 − 1×2)/(2 − 1) )
- P = ( (10 − 1)/1, (12 − 2)/1 )
- P = (9, 10)
5. Why do we use m − n in the external division formula?
We use (m − n) in the denominator because the point lies outside the line segment, creating a difference of weighted distances. In external division, the distances are measured in opposite directions, which leads to subtraction instead of addition as in internal division. This ensures the point lies beyond either A or B.
6. When is the external division formula undefined?
The external division formula is undefined when m = n because the denominator (m − n) becomes zero. Division by zero is not defined in mathematics. Therefore, external division is possible only when m and n are unequal.
7. How do you know if a point divides a line internally or externally?
A point divides a line internally if it lies between the two endpoints, and externally if it lies outside the segment.
- If the formula uses (m + n), it is internal division.
- If the formula uses (m − n), it is external division.
- External division often results in coordinates beyond the range of A and B.
8. What are the applications of the external division formula?
The external division formula is used to find points outside a line segment in coordinate geometry problems.
- Finding points of intersection in algebraic proofs
- Solving section formula problems in exams
- Coordinate geometry constructions
- Vector and analytic geometry applications
9. Can the external division formula be used in three dimensions?
Yes, the external division formula can be extended to 3D coordinate geometry. If A(x₁, y₁, z₁) and B(x₂, y₂, z₂) are divided externally in the ratio m:n, then:
P = ( (mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n), (mz₂ − nz₁)/(m − n) )
This formula works similarly to the 2D case but includes the z-coordinate.
10. What are common mistakes when using the external division formula?
Common mistakes in the external division formula usually involve sign errors and denominator confusion.
- Using (m + n) instead of (m − n)
- Substituting coordinates in the wrong order
- Forgetting that m ≠ n
- Making arithmetic mistakes while simplifying
