Minimum Distance from a Point to a LineDate: 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 |
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