Question

What is the distance between lines 3x − 5y + 3=0 and 6x − 10y − 12=0?

Answer 1

Let us rewrite the two lines in slope intercept form to obtain,

`3x-5y+3=0`

`\Rightarrow -5y=-3x-3`

`\Rightarrow y=(3)/(5)x+(3)/(5)`

For the second equation too, we have,

`6x-10y-12=0`

`\Rightarrow -10y=-6x+12`

`\Rightarrow y=(6)/(10)x-(12)/(10)`

`\Rightarrow y=(3)/(5)x-(6)/(5)`

The two slopes are equal. this means the two lines are parallel. We apply the formula for finding the distance between 2 parallel lines. This formula is given by

`d=(|b_2-b_1|)/(√(m^2+1) )`

Where

`b_1=(3)/(5)`

`b_2=-(6)/(5)`

and

`m=(3)/(5)`

We substitute into the formula to obtain,

`d=\frac{\sqrt{((3)/(5))^2 +1 } }`

`d=\frac{\sqrt{(9)/(25) +1}}`

`d=\frac{\sqrt{(34)/(25) } }`

`d=(9√(34) )/(34)`

Therefore the distance between the two lines is approximately 1.5 units.