Question

What is the shortest distance between two parallel lines?

Answer 1

How do I find distance between two given equations of line?

3 ANSWERS



Kumar Pushpesh, likes solving questions

Answered Oct 1, 2016

I’m assuming that you’re concerned with 22D geometry here.

To find the distance between 22 lines, first of all you must understand that a unique distance is possible only when the lines are parallel. If not, then the distance between them will vary. We should also note that the word distance here implies shortest distance or perpendicular distance.

General case of parallel lines ( equations )

a1x+b1y+c1=0a1x+b1y+c1=0a2x+b2y+c2=0a2x+b2y+c2=0

Now, since the lines are parallel : a1a2=b1b2=ka1a2=b1b2=k

After some basic manipulation, we can convert the second equation as

a1x+b1y+c3=0a1x+b1y+c3=0

where c3=kc2c3=kc2

The required distance can be calculated as

d=|c3−c1|a21+b21−−−−−−√d=|c3−c1|a12+b12

For proof take any point on one of the lines and from that point find the equation of a line perpendicular to the line. This line intersects the other line at some point. You can use the standard distance formula between 2 points to derive the distance required.

Let me know if there’s any issue.