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June 23rd, 2012, 04:48 PM   #1
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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.
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June 23rd, 2012, 09:02 PM   #2
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Re: Finding The Shortest Distance Between Two Points...

Let point P be and point Q be and using the two-intercept 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:



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June 24th, 2012, 08:11 AM   #3
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Re: Finding The Shortest Distance Between Two Points...

thanks mark going through your answer now!
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June 24th, 2012, 01:16 PM   #4
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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?
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June 24th, 2012, 01:55 PM   #5
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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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File Type: jpg lengthsquared.jpg (9.4 KB, 200 views)
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