## About "How to find the ratio in which a point divides a line"

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 [ (lx2 + mx1)/(l + m),  (ly2 + my1)/(l + m) ]  =  (-2, 3)

(x1, y1) ==>  (-3, 5) and (x2, y2) ==>  (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".

If you want to know more about the stuff "How to find the ratio in which a point divides a linePlease click here

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6