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There exists a constant c so that among all chords  of the parabola  passing through 
is a fixed constant. Find the constant 

 Feb 28, 2020
 #1
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To get a handle on the constant  we can look at some particular cases.

Suppose we let  approach a vertical line. Then  approaches 0, and  approaches  so  approaches  Hence,
Now, suppose we take  and  Then
Hence,  so  and  Therefore,  (Note that this makes  the focus of the parabola.)

For a full solution, let's check that this value works. Let  be the equation of line  Setting  we get
or  Let  and  be the roots of this equation. By Vieta's formulas,  and 

Also,  and  are  and  in some order, so
We have that  and
Hence,
To get a handle on the constant  we can look at some particular cases.

Suppose we let  approach a vertical line. Then  approaches 0, and  approaches  so  approaches  Hence,
Now, suppose we take  and  Then
Hence,  so  and  Therefore,  (Note that this makes  the focus of the parabola.)

For a full solution, let's check that this value works. Let  be the equation of line  Setting  we get
or  Let  and  be the roots of this equation. By Vieta's formulas,  and 

Also,  and  are  and  in some order, so
We have that  and
Hence,
 

 Feb 28, 2020

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