Question

P ( a, b ) is the mid-point of a line segment between axes. Show that equation of the line is

Answer 1

The point P( a,b ) is the mid-point of line segment between the axes.

Suppose the coordinate on x axis is ( x ′ ,0 ) and on y axis is ( 0, y ′ ) respectively.

The formula for the coordinates of mid-point ( x m , y m ) of a line segment joining the points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by

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

Substitute the values of ( x m , y m ) , ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( a,b ) , ( x ′ ,0 ) and ( 0, y ′ ) respectively in equation(1).

( a,b )=( 0+ x ′ 2 , 0+ y ′ 2 ) =( x ′ 2 , y ′ 2 )

Compare the values of above expression on both hand sides.

x ′ 2 =a    y ′ 2 =b x ′ =2a    y ′ =2b

The coordinates of the points on x axis and y axis is given as ( 2a,0 ) and ( 0,2b ) .

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 ( 2a,0 ) and ( 0,2b ) respectively in equation (2).

( y−2b )= 0−2b 2a−0 ⋅( x−0 ) ( y−2b )= −2b 2a ⋅( x−0 ) ( y−2b )= −b a ⋅( x−0 ) a⋅( y−2b )=−b⋅( x−0 )

Further simplify the above equation.

ay−2ab=−bx bx+ay−2ab=0 bx+ay=2ab

Divide both side by ab .

bx ab + ay ab = 2ab ab x a + y b =2

Hence the equation of line segment between axes with mid-point of line segment P( a,b ) is x a + y b =2 .