Question

Point R ( h, k ) divides a line segment between the axes in the ratio 1:2. Find equation of the line.

Answer 1

The point R ( h,k ) divides the line segment between the axes in the ratio of 1:2 .

Let the coordinates on x axis is ( x ′ ,0 ) and on y axis is ( 0, y ′ ) respectively.

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 ) (1)

Substitute the values of the points ( x z , y z ) , ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( h,k ) , ( x ′ ,0 ) , and ( 0, y ′ ) respectively in equation (1).

Substitute the value of m:n as 1:2 in equation (1).

( h,k )=( 1×0+2× x ′ 1+2 , 1× y ′ +2×0 1+2 ) =( 2 x ′ 3 , y ′ 3 )

Compare the values on both hand sides.

2 x ′ 3 =h    y ′ 3 =k x ′ = 3h 2     y ′ =3k

The coordinates of the points on x axis and y axis is given as ( 3h 2 ,0 ) and ( 0,3k ) .

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 ) (2)

Substitute the values of ( x 1 , y 1 ) , ( x 2 , y 2 ) as ( 3h 2 ,0 ) and ( 0,3k ) . respectively in equation (2).

( y−0 )= 3k−0 0− 3h 2 ⋅( x− 3h 2 ) ( y−0 )= 3k −3h 2 ⋅( x− 3h 2 ) ( y−0 )= −6k 3h ⋅( x− 3h 2 ) ( y−0 )= −2k h ⋅( x− 3h 2 )

Further simplify the above expression.

h⋅( y−0 )=−2k⋅( x− 3h 2 ) hy=−2kx+2k⋅ 3h 2 hy=−2kx+k⋅3h 2kx+hy−3hk=0

Thus the equation of line segment divided by point R ( h,k ) in the ratio 1:2 is 2kx+hy−3hk=0 .