Question

Prove that the line through the point ( x 1 , y 1 ) and parallel to the line A x + B y + C = 0 is A ( x –x 1 ) + B ( y – y 1 ) = 0.

Answer 1

The line passes through the point ( x 1 , y 1 ) and parallel to the line Ax+By+C=0 .

The equation of line having slope m and making an intercept c with the y axis is given by

y=mx+c (1)

Let m 1 is the slope of the line Ax+By+C=0 .

Rearrange the terms of the equation of line 7x−9y−19=0 .

By=−Ax−C y=( − A B )x+( − C B )

Compare the above equation with the equation (1).

m 1 =− A B

Let m 2 is the slope of the line that passes through the point ( x 1 , y 1 ) .

As both the lines are parallel to each other,

m 1 = m 2 m 2 =− A B

The formula for the equation of the line having slope m passes through the point ( x 1 , y 1 ) is given by,

( y− y 1 )=m( x− x 1 ) (2)

Substitute the values of m as − A B in equation (2).

( y− y 1 )=− A B ⋅( x− x 1 ) B⋅( y− y 1 )=−A⋅( x− x 1 ) A⋅( x− x 1 )+B⋅( y− y 1 )=0 A( x− x 1 )+B( y− y 1 )=0

Hence the equation of line passes through the point ( x 1 , y 1 ) and parallel to the line Ax+By+C=0 is A( x− x 1 )+B( y− y 1 )=0 .