We use the section formula to find the point which divides the line segment in a given ratio.

The point P which divides the line segment joining the two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}) internally in the ratio l : m is

If P divides a line segment AB joining the two points

A (x_{1}, y_{1}) and B (x_{2}, y_{2}) externally in the ratio l : m is,

**Example 1 :**

Find the coordinates of the point which divides the line segment joining (-3, 5) and (4, -9) in the ratio 1:6 internally.

**Solution :**

Let A (-3, 5) and B (4, -9)

Section formula internally

= (lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)

l = 1 and m = 6

= [(1(4)+(6(-3)]/(1+6) , [(1(-9)) + 6(5)]/(1+6)

= (4-18)/7, (-9 + 30)/7

= -14/7, 21/7

= (-2, 3)

**Example 2 :**

Let A (-6 , -5) and B(-6 , 4) be the 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

9AP = 2(AP+PB)

9AP = 2AP + 2PB

9AP – 2AP = 2PB

7AP = 2PB

AP/AB = 2/7

AP:PB = 2:7

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

Section formula internally

= (lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)

l = 2, m = 7

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

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

= -54/9, -21/7

= (-6, -3)

**Example 3 :**

Find the points of trisection of the line segment joining the points A (2, -2) and B (-7, 4).

**Solution :**

AP = 1, PQ = 1, QB = 1

= (lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)

P divides the line segment in the ratio 1:2

l = 1, m = 2

= [(1(-7)+2(2)]/(1+2), [1(4)+2(-2)]/(1+2)

= (-7+4)/3, (4-4)/3

= -3/3 , 0/3

= P (-1 , 0)

Q divides the line segment in the ratio 2:1

l = 2, m = 1

= [2(-7)+1(2)]/(2+1), [2(4)+1(-2)]/(2+1)

= (-14+2)/3, (8-2)/3

= -12/3 , 6/3

= Q (-4, 2)

**Example 4 :**

Find the ratio in which 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

Section formula internally =

(lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)

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

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

Equating y-coordinates

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

-7l+4m = 0

-7l = -4m

l/m = 4/7

l : m = 4 : 7

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

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