# Section Formula

**Internal Division**

Let P_{1}(x_{1},y_{1}) and P_{2}(x_{2},y_{2}) be two points. Let P(x, y) be a point on the segment joining P_{1} and P_{2} such that P divides the line P_{1 }, P_{2} in the ratio m_{1}: m_{2}, where m_{1}, m_{2} are positive real numbers. Then

Draw lines P_{1}M_{1}, PM and P_{2}M_{2} parallel to the y-axis to meet the x-axis in M_{1}, M and M_{2} as shown in figure. We have

OM_{1} = -x_{1}, M_{1}P_{1} = y_{1}, OM = -x, PM = y, OM_{2} = x_{2}, P_{2} M_{2} = y_{2 }

Draw P_{1}Q_{1} and PQ parallel to OX, to meet MP and M_{2}P_{2} in Q_{1} and Q, respectively. Then

P_{1}Q_{1} = M_{1}M = OM_{1} â€“ OM = x âˆ’ x_{1 }

PQ = MM_{2} = MO + OM_{2} = x_{2} âˆ’ x

Q_{1}P = PM âˆ’ MQ_{1} = y âˆ’ y_{1 }

QP_{2} = M_{2}p_{2} âˆ’ M_{2}Q = y_{2} âˆ’ y

From the similar triangles P_{1}Q_{1} P and PQP_{2}, we have

Therefore, m_{1} (x_{2 } âˆ’ x) = m_{2} (x âˆ’ x_{1}). 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(x_{1},y_{1}) and Q(x_{2},y_{2}), then m_{1} = m_{2} 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 P_{1}(x_{1},y_{2}) and P_{2}(x_{2},y_{2}) in the ratio m_{1}: m_{2}.

**Step 1 **

Multiply m_{1} with the x-coordinate of P_{2 }and m_{2} with the x-coordinate of P_{1}.

**Step 2 **

Add the products obtained in step 1.

**Step 3 **

Divide the sum in Step 2 by m_{1}+ m_{2} 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 m_{1} : m_{2}). 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 P_{1}(x

_{1},y

_{1}) and P

_{2}(x

_{2},y

_{2}) divides the segment in some ratio, its coordinates are given by

kâ‰ -1.

Thus, x = (kâ‰ -1).

represent the parametric coordinates of all points lying on the line P_{1}P_{2}.

**Short Answers **

Example

(i) Find the coordinates of the point which divides the line joining the points (5, âˆ’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 (âˆ’1, 3) and (4, âˆ’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 âˆ’ x + 2 = 0 divides the line joining (3, âˆ’1) and (8, 9).

**Solution**

(i) The coordinates of the point which divides the line joining the points (5, âˆ’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 (âˆ’6, âˆ’4).

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

Thus, the coordinates of the required point are (1, âˆ’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

â‡’ 16k + 11 = 9k + 9

â‡’ 7k = âˆ’2

â‡’ k =

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

(v) Suppose 2y âˆ’ x + 2 = 0 divides the line joining the points A(3, âˆ’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 â€“ x + 2 = 0, we get

â‡’ 18k â€“ 2 â€“ 8k â€“ 3 + 2k + 2 = 0

â‡’ 12k âˆ’ 3 = 0

â‡’ k = .

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