Coordinate Geometry

Question

The line y=2x+7 is parallel to the line y=2x-3.The line y=-0.5x+7 is perpendicular to the line y=2x-3.Calculate the perpendicular distance between the pair of parallel lines.

Answer 1

y = 2x + 7    parallel to   y = 2x-3

 

Since y= - 0.5x+7 is perpendicular to y=2x-3, and y= 2x-3 is parallel to y = 2x+7,

then y = -0.5x is also perpendicular to y = 2x+7

 

To find the perpendicular distance between the parallel lines we need to find

where y = 2x+7 crosses y=-0.5x+7

and where y = 2x-3 crosses y = -0.5x+7

The distance between those two points will be the perpendicular distance

between the two parallel lines

 

Let's work on the solution of y = 2x + 7 and y = -0.5x +7

 

y =    2x+7            eqn 1

y = -0.5x + 7         eqn 2

-----------------

 

y = 2x + 7          eqn 1

4y=-2x+28         eqn 2(4)

------------

5y = 35

y = 7

 

Substitute y = 7 back into eqn 1 giving

7 = 2x + 7

2x = 0

x = 0

 

Line y = 2x + 7 meets y = -0.5x + 7 at the point (0,7)

**********************

Let's work on the intersection of y = 2x-3 and y = -0.5x + 7

 

y =   2x - 3       eqn 1

y = -0.5x + 7    eqn 2

---------------

 

y =  2x - 3          eqn 1

4y = -2x + 28     4(eqn 2)

----------------

5y = 25

y = 5

 

Substitute y = 5 back into y = 2x-3

5 = 2x-3

2x=8

x=4

 

Line y = -0.5x + 7 intersects y = 2x-3 at the point (4,5)

 

The perpendicular distance between the parallel lines is the

distance from (0,7) to (4,5)

 

d = √((x₂-x₁)²+(y₂-y₁)²)

d = √((4-0)²+(5-7)²)

d = √((4²) + (-2)²)

d = √(16+4)

d = √20

d = 2√5

 

The perpendicular distance between the two parallel lines is 2√5