Minimum Distance from a Point to a Line

Date: 10 Jul 1995 14:13:31 -0400
From: Margie Salaz
Subject: Circles and lines

I am having trouble figuring out the following problem.  Could you explain 
it to me?

Find all the values of b such that the minimum distance from the point (2,0) 
to the line y = 4/3x+b is 5.

Thank you so much for your help. I greatly appreciate it.

Sincerely,

Margie Salaz
Cuba, NM

Date: 12 Jul 1995 22:07:16 -0400
From: Dr. Ken
Subject: Re: Circles and lines

Hello there!

When you say "minimum distance from the point (2,0)," the buzzer that should
go off in your head is "hey, all the places that are 5 units away from (2,0)
lie on a circle of radius 5!"  That's the first realization that will go
really far.  The second is that whenever you're talking about the minimum
distance to a line, you're talking about the distance along a line
perpendicular to a given line (sometimes called the "perpendicular
distance").  Since tangents to a circle are always perpendicular to the
radii of the circle, we're looking for lines of the form 4/3x+b that are
tangent to the circle of radius 5 centered at (2,0).  Essentially, what that
means is find all lines with slope 4/3 that are tangent to that circle.

With that said, draw a picture.  That's really the best thing to do.  How
many such lines can there be?  How far apart are they?  Can you find their
equations?  If you need more help than this hint, write back and we'll help
you out.

-K