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June 23rd, 2012, 04:48 PM  #1 
Senior Member Joined: Jan 2012 Posts: 159 Thanks: 0  Finding The Shortest Distance Between Two Points...
that also is tangent to a given function. No idea about this one. Seems not that tough just do not know where to begin. 
June 23rd, 2012, 09:02 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Finding The Shortest Distance Between Two Points...
Let point P be and point Q be and using the twointercept formula for a line, the line passing through P and Q is given by: Now substitute for y using : Multiply through by x and arrange in standard quadratic form: We require this equation to have 1 real repeated root, hence the discriminant must be zero: Now, the square of the distance L from P to Q is: Implicitly differentiate with respect to : Simplify and equate to zero: Hence: 
June 24th, 2012, 08:11 AM  #3 
Senior Member Joined: Jan 2012 Posts: 159 Thanks: 0  Re: Finding The Shortest Distance Between Two Points...
thanks mark going through your answer now!

June 24th, 2012, 01:16 PM  #4 
Senior Member Joined: Jan 2012 Posts: 159 Thanks: 0  Re: Finding The Shortest Distance Between Two Points...
I am surprised I am actually understanding your solution. Just did a review of the what you can tell from the value of a discriminant, cool stuff. So I am down to where you set dL/xp equal to zero. This is the rate of change of the distance over the rate of change of xp. So I know we are looking for the min L (distance) but could you explain a bit why we set this ratio to 0? 
June 24th, 2012, 01:55 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Finding The Shortest Distance Between Two Points...
Here is a graph of . This should make it obvious why we set the derivative equal to zero: [attachment=0:37i9uwqm]lengthsquared.jpg[/attachment:37i9uwqm] 

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