Question

A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n . Find the equation of the line.

Answer 1

The line segment is formed by joining the points ( 1,0 ) and ( 2,3 ) . Also the line perpendicular to this line segment divide it in ratio of 1:n .

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)

Let the coordinates of the point dividing the line segment is ( p,q ) .

Substitute the values of ( x z , y z ) , ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( p,q ) , ( 1,0 ) , ( 2,3 ) and m:n as 1:n respectively in equation (1).

( p,q )=( 1⋅2+n⋅1 1+n , 1⋅3+n⋅0 1+n ) =( 2+n 1+n , 0+3 1+n ) =( 2+n n+1 , 3 n+1 ) (2)

Thus, the point ( p,q ) lies on the line segment.

Now, the formula for the slope of a line passes through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (3)

Let m 1 be the slope of the line segment which passes through the points ( 1,0 ) , ( 2,3 ) .

Substitute the value for ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 1,0 ) , ( 2,3 ) respectively in equation (3).

m 1 = 3−0 2−1 = 3 1 =3 (4)

When the two vertical lines are perpendicular to each other the product of their slopes are equal to -1.

m 1 ⋅ m 2 =−1 (5)

Where m 2 be the slope of the perpendicular line through point ( p,q ) .

Substitute the value of m 1 from equation (4) to equation (5).

3⋅ m 2 =−1 m 2 = −1 3

The formula for the equation of a non-vertical line having slope m and passing through the point ( x 0 , y 0 ) is given by,

( y− y 0 )=m( x− x 0 ) (6)

Substitute the value of ( x 0 , y 0 ) as ( p,q ) from equation (2) and m as −1 3 .in equation (6).

( y− 3 n+1 )= −1 3 ⋅( x− n+2 n+1 ) 3⋅( y− 3 n+1 )=−1⋅( x− n+2 n+1 ) 3y− 9 n+1 =−x+ n+2 n+1 x−( n+2 n+1 )+3y− 9 n+1 =0

Take ( n+1 ) as common L.C.M.

x( n+1 ) n+1 −( n+2 n+1 )+ 3y( n+1 ) n+1 − 9 n+1 =0 x( n+1 )−( n+2 )+3y( n+1 )−9=0 ( n+1 )x+3( n+1 )y−9−n−2=0 ( n+1 )x+3( n+1 )y−n−11=0

Thus, the equation of line perpendicular to line segment formed by joining the points ( 1,0 ) and ( 2,3 ) and divide it in ratio of 1:n is ( n+1 )x+3( n+1 )y−n−11=0 .