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Section Formula

Internal Division

Let P1(x1,y1) and P2(x2,y2) be two points. Let P(x, y) be a point on the segment joining P1 and P2 such that P divides the line P1 , P2 in the ratio m1: m2, where m1, m2 are positive real numbers. Then

Draw lines P1M1, PM and P2M2 parallel to the y-axis to meet the x-axis in M1, M and M2 as shown in figure. We have
 


OM1 = -x1, M1P1 = y1, OM = -x, PM = y, OM2 = x2, P2 M2 = y2

Draw P1Q1 and PQ parallel to OX, to meet MP and M2P2 in Q1 and Q, respectively. Then

P1Q1 = M1M = OM1 – OM = x − x1

PQ = MM2 = MO + OM2 = x2 − x

Q1P = PM − MQ1 = y − y1

QP2 = M2p2 − M2Q = y2 − y

From the similar triangles P1Q1 P and PQP2, we have

Therefore, m1 (x2 − x) = m2 (x − x1). This gives x =

Similarly,

In this case the coordinates of P are given by

                             

Mid-point Formula
If P is the mid-point of the segment P(x1,y1) and Q(x2,y2), then m1 = m2 and the coordinates of P are given by  

 

Rules to Write Down the Coordinates of a Point Dividing a Segment in given ratio

Rule for Internal Division

 

To divide the line joining the points P1(x1,y2) and P2(x2,y2) in the ratio m1: m2.


Step 1
Multiply m1 with the x-coordinate of P2 and m2 with the x-coordinate of P1.

 

 

Step 2
Add the products obtained in step 1.


Step 3
Divide the sum in Step 2 by m1+ m2 and put it equal to x.

(This gives x-coordinate of point P)


Step 4
Repeat Step 1 to Step 3 after replacing x by y.

(This gives y-coordinate of point P).

Finding the Ratio of a Point Which Divides The Line Joining Two Given Points.
In case we are interested to find out a ratio, we take it to be k : 1 (rather than as m1 : m2). Note that we now have to find merely one constant k rather than two.



Parametric Representation of straight line through P1 and P2

Since every point P lying on the line joining the points P1(x1,y1) and P2(x2,y2) divides the segment in some ratio, its coordinates are given by

 

k≠ -1.

Thus, x = (k≠ -1).

represent the parametric coordinates of all points lying on the line P1P2.

Short Answers

Example
(i) Find the coordinates of the point which divides the line joining the points (5, −2) and (9, 6) internally in the ratio 1: 3.


(ii) Find the coordinates of the point which divides the line joining the points (2, 4) and (6, 8) in the ratio 2 : 3 internally and externally.


(iii) Find the coordinates of the points which divide the line joining (−1, 3) and (4, −7) internally in the ratio 2 : 3.


(iv) In what ratio is the line joining (2, 5) and (3, 7) divided by 3x + y = 9?


(v) Determine the ratio in which 2y − x + 2 = 0 divides the line joining (3, −1) and (8, 9).


Solution
(i) The coordinates of the point which divides the line joining the points (5, −2) and (9, 6) internally in the ratio 1 : 3 are
 

 

Thus the coordinates of the required point are (6, 0).

 

(ii) The coordinates of the point which divides the line joining the points (2, 4) and (6, 8) internally in the ratio of 2 : 3 are






Thus coordinates of the required point are .

The coordinates of the point which divides the line joining the points (2, 4) and (6, 8) externally in the ratio of 2 : 3 are


Thus, the coordinates of the required point are (−6, −4).


(iii) The coordinates of the point which divides the line joining the points (−1, 3) and (4, −7) internally in the ratio 2 : 3 are given by
 

 

Thus, the coordinates of the required point are (1, −1).


(iv) Suppose 3x + y = 9 divides the line joining the points A(2, 5) and B(3, 7) in the ratio k : 1.

Let the point of intersection be C. The coordinates of C are given by

Since C lies on the line 3x + y = 9, we have

⇒ 16k + 11 = 9k + 9

⇒ 7k = −2

⇒  k =

Thus, 3x + y = 9 divides the line joining the points A(2, 5) and B(3, 7) in the ratio : 1 or –2 : 7. Hence, 3x + y = 9 divides the line joining the points A and B externally in the ratio 2 : −7.

 

(v) Suppose 2y − x + 2 = 0 divides the line joining the points A(3, −1) and B(8, 9) in the ratio k:1.

Let the point of intersection be C. The coordinates of C are given by . Since C lies on the line 2y – x + 2 = 0, we get

⇒ 18k – 2 – 8k – 3 + 2k + 2 = 0

⇒ 12k − 3 = 0

⇒ k = .

Thus, 2y – x + 2 = 0 divides the line joining the points A(3, −1) and B(8, 9) in the ratio : 1 or 1 : 4.
 





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