Question

Find the ratio in which the line joining the points(2,3) and (4,1) divides the segment joining points (1,2) and (4,3)​

Answer 1

Given :

Line segments with points (2,3) and (4,1) & points (1,2) and (4,3)​

To Find : The ratio in which line segment joining point (2,3) and (4,1) divides the segment joining points (1,2) and (4,3)​

Solution :

•To Find point of intersection of two line segments we need to find the equation of lines

•slope of line segment passing from

(2,3) and (4,1)

m = (y2-y1)/(x2-x1)

m = (1-3)/(4-2) = -2/2 = -1

•Equation of line segment passing from (2,3) and (4,1)

y - y1 = m ( x - x1 )

y - 3 = (-1) ( x - 2)

y - 3 = -x + 2

x + y = 5 _____(1)

•slope of line segment passing from (1,2) and (4,3)​

m = (y2-y1)/(x2-x1)

m = (3-2)/(4-1) = 1/3

•Equation of line segment passing from (1,2) and (4,3)​

y - y1 = m ( x - x1 )

y - 2 = (1/3) ( x - 1)

3(y - 2) = x - 1

3y - 6 = x - 1

3y - x = 5 _______(2)

•Now , both the lines intersect at point ( 5/2 , 5/2 )

•let the points ( 5/2 , 5/2 ) divides line segment joining points (1,2) and (4,3)​ in ratio k:1

•By section formula ,

5/2 = [ k(4) + (1)(1) ]/(k+1)

5(k+1) = 2(4k + 1)

5k + 5 = 8k + 2

3k = 3

k = 1

•Hence , line segment joining point (2,3) and (4,1) divides the segment joining points (1,2) and (4,3)​ in the ratio 1:1

Answer 2

Given :

The point of line AB

A = 2 , 3

B = 4 , 1

The line AB divide line PQ

P = 1 ,2

Q = 4 , 3

To Find :

The ratio in which  line AB divides line PQ

Solution :

Equation of line AB

 y - 3 = ( $\dfrac{1-3}{4-2}$ ) ( x - 2 )

Or, y - 3 = ( $\dfrac{-2}{2}$ ) ( x - 2 )

Or,   y - 3 = ( -1 ) ( x - 2 )

∴    y - 3 = - x + 2

i,e  y = - x + 5   is eq of line AB              ...........1

Again

Equation of line PQ

 y - 2 = ( $\dfrac{3-2}{4-1}$ ) ( x - 1 )

Or, y - 2 = ( $\dfrac{1}{3}$ ) ( x - 1 )

Or,  3 y - 6 = ( 1 ) ( x -1 )

∴    3 y =  x + 5       is eq of line PQ                   ............2

Solving eq 1 and eq 2

  i.e  3 (  - x + 5 ) = x + 5

Or,  - 3 x + 15 = x + 5

Or,  x + 3 x = 15 - 5

Or,        4 x = 10

∴             x = $\dfrac{5}{2}$

Now, Put the value of x into eq 1

i.e  y = - x + 5

       = - $\dfrac{5}{2}$ + 5

       = $\dfrac{-5+10}{2}$ = $\dfrac{5}{2}$

So, The point of intersection of line AB and PQ = $\dfrac{5}{2}$ , $\dfrac{5}{2}$

Let, The ratio in which both divides = m : 1

So,   $\dfrac{m\times 4+1\times 1}{m+1}$ ,  $\dfrac{m\times 3+1\times 2}{m+1}$

Or,   $\dfrac{4m+1}{m+1}$ ,  $\dfrac{3m+2}{m+1}$

∴  $\dfrac{4m+1}{m+1}$ ,  $\dfrac{3m+2}{m+1}$  =  $\dfrac{5}{2}$ , $\dfrac{5}{2}$

i.e  $\dfrac{4m+1}{m+1}$  =  $\dfrac{5}{2}$

By cross multiplication

2 ( 4 m + 1 ) = 5 ( m + 1 )

Or,  8 m + 2 = 5 m + 5

Or, 8 m - 5 m = 5 - 2

Or,          3 m = 3

∴               m = 1

Similarly

 $\dfrac{3m+2}{m+1}$  =  $\dfrac{5}{2}$

By cross multiplication

2 ( 3 m + 2 ) = 5 ( m + 1 )

Or, 6 m + 4 = 5 m + 5

Or,  6 m - 5 m = 5 - 4

∴ m = 1

Thus, ratio in which lines divides = m : 1 = 1 : 1

Hence, The ratio in which line joining points divides segments points in 1 : 1  Answer