**How to find the ratio in which a point divides a line :**

Here we are going to see how to find the ratio in which a point divides the line.

Let us look into some examples to understand the above concept.

**Example 1 :**

In what ratio does the point P(-2 , 3) divide the line segment joining the points A(-3, 5) and B (4, -9) internally?

**Solution :**

Given points are (-3 , 5) and B (4 ,- 9).

Let P (-2 , 3) divide AB internally in the ratio l : m By the section formula,

P [ (lx_{2} + mx_{1})/(l + m), (ly_{2} + my_{1})/(l + m) ] = (-2, 3)

(x_{1}, y_{1}) ==> (-3, 5) and (x_{2}, y_{2}) ==> (4, -9)

(l(4) + m(-3))/(l+m) , (l(-9) + m(5))/(l+m) = (-2, 3)

(4l - 3m)/(l+m), (-9l + 5m)/(l+m) = (-2, 3)

Equating the coefficients of x, we get

(4l - 3m)/(l+m) = -2

4l - 3m = -2(l + m)

4l - 3m = -2l - 2m

Add 2l and 3m on both sides

6l = m

l/m = 1/6

l : m = 1 : 6

Hence the point P divides the line segment joining the points in the ratio 1 : 6.

**Example 2 :**

Let A (-6,-5) and B (-6, 4) be two points such that a point P on the line AB satisfies AP = (2/9) AB. Find the point P.

**Solution :**

AP = (2/9) AB

9 AP = 2 (AP + PB)

9 AP = 2 AP + 2 PB

9 AP – 2 AP = 2 PB

7 AP = 2 PB

AP/AB = 2/7

AP: PB = 2: 7

So P divides the line segment in the ratio 2:7

Section formula internally

= (lx₂ + mx₁)/(l + m) , (ly₂ + my₁)/(l + m)

L = 2 m = 7

= [(2(-6)) + (7(-6)]/(2+7) , [(2(4)) + (7(-5)]/(2+7)

= (-12-42)/9 , (8 - 35)/9

= (-54/9 , -21/7)

= (-6 , -3)

**Example 3 :**

Find the ratio in which the x-axis divides the line segment joining the points (6, 4) and (1,-7).

Solution :

Let L : m be the ratio of the line segment joining the points (6,4) and (1,-7) and let p(x,0) be the point on the x axis

= (Lx₂ + mx₁)/(L + m) , (Ly₂ + my₁)/(L + m)

(x, 0) = [l(1) + m(6)]/(l + m) , [l(-7) + m(4)]/(l + m)

(x, 0) = [L + 6 m]/(l + m) , [-7l + 4m]/(l + m)

Equating y-coordinates, we get

[-7l + 4m]/(L + m) = 0

- 7 l + 4 m = 0

- 7 l = - 4 m

l/m = 4/7

l : m = 4 : 7

Hence x-axis divides the line segment in the ratio 4 : 7.

Let us see the next example on "How to find the ratio in which a point divides a line".

**Example 4 :**

In what ratio is the line joining the points (-5 ,1) and (2 ,3) divided by y-axis? Also, find the point of intersection.

**Solution :**

Let L : m be the ratio of the line segment joining the points (-5 , 1) and (2 ,3) and let p(0,y) be the point on the y axis

Section formula internally

= (lx2 + mx1)/(l + m) , (ly2 + my1)/(l + m)

(0 , y) = [L(2) + m(-5)]/(L + m) , [L(3) + m(1)]/(L + m)

(0 , y) = [2L - 5 m]/(L + m) , [3L + m]/(L + m)

[2L - 5 m]/(L + m) = 0

2 l - 5 m = 0

2 l = 5 m

l/m = 5/2

l : m = 5 : 2

To find the required point we have to apply this ratio in the formula

(0 , y) = [2(5) – 5(2)]/(5 + 2) , [3(5) + 2]/(5 + 2)

(0 , y) = [10 – 10]/7 , [15 + 2]/7

(0 , y) = (0 , 17/7)

Hence the required point is (0,17/7)

After having gone through the stuff given above, we hope that the students would have understood "How to find the ratio in which a point divides a line".

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