Question

Find the distance of the line 4 x + 7 y + 5 = 0 from the point (1, 2) along the line 2 x – y = 0.

Answer 1

The equation of lines is

4x+7y+5=0(1)

2x−y=0(2)

According to the question, the point ( 1,2 ) lies on equation (2).

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

Substitute the value of y=2x from equation (2) to equation (1).

4x+7( 2x )+5=0 4x+14x+5=0 18x+5=0 x=− 5 18

Substitute the value of x in equation (2) to obtain the value of y.

y=2×( −5 18 ) =− 5 9

The coordinates of point of intersection are ( −5 18 , −5 9 ).

The distance has to be found along the line.

The formula for the distance between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

d= ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 1,2 ) and ( −5 18 , −5 9 ) in the above expression.

d= ( 1+ 5 18 ) 2 + ( 2+ 5 9 ) 2 = ( 23 18 ) 2 + ( 23 9 ) 2 = ( 23 2×9 ) 2 + ( 23 9 ) 2 = ( 23 9 ) 2 ( 1 4 +1 )

Further simplify the above expression.

d=( 23 9 ) 5 4 = 23 9×2 5 = 23 18 5  units

Thus, the distance of the point ( 1,2 ) along the line 4x+7y+5=0 and along the line 2x−y=0 is 23 18 5  units.