Question

In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?

Answer 1

The line is passing through the points ( −1,1 ) and ( 5,7 ).

The formula for the equation of line passing through the points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

( y− y 1 )= y 2 − y 1 x 2 − x 1 ⋅( x− x 1 )(1)

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( −1,1 ) and ( 5,7 ) in equation (1).

( y−1 )= 7−1 5+1 ⋅( x+1 ) y−1= 6 6 ⋅( x+1 ) y−1=x+1 x−y+2=0

As per question, the line x−y+2=0 divides the line x+y=4.

The point of intersection of these two lines is obtained by solving the two equations of line.

Substitute the value of y=x+2 in equation of line x+y=4.

x+x+2=4 2x=2 x=1

Substitute the value of x in equation of line to obtain y.

y=3.

The coordinates of the intersection point are ( 1,3 ).

Let this point divide the line segment x−y+2=0 in the ratio ( 1:k ) .

The formula for the coordinates of a point ( x z , y z ) dividing the line segment joining the points ( x 1 , y 1 ) and ( x 2 , y 2 ) internally in a ratio of m:n is given by,

( x z , y z )=( m x 2 +n x 1 m+n , m y 2 +n y 1 m+n )(2)

Substitute the value of ( x z , y z ) as ( 1,3 ), ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( −1,1 ) and ( 5,7 ), and m:n as 1:k respectively in equation (2).

( 1,3 )=( 1×5+k×( −1 ) k+1 , 1×7+k×1 k+1 ) ( 1,3 )=( −k+5 k+1 , k+7 k+1 )

Compare the values on both the sides.

−k+5 k+1 =1 −k+5=k+1 2k=4 k=2

The ratio is ( 1:2 ).

Thus, the line is passing through the points ( −1,1 ) and ( 5,7 ) divides line x+y=4 in the ratio ( 1:2 ).