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Determine the coordinates of the point \(P\) on the line \(y=-x+6\) such that \(P\) is equidistant from the points \(A(10,-10)\) and \(O(0,0)\) (that is, so that \(PA=PO\)). Express your answer as an ordered pair \((a,b)\).

 Jun 11, 2019
 #1
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May be several ways to do this

 

Notice that the slope of a segment drawn between O and A  =  [ -10- 0] /[10-0]  = -10 /10 = -1

 

Which is the same slope of the line y = -x + 6

 

We can find the midpoint of OA  and write an equation of the line through this midpoint that is perpendicular to y = - x + 6

 

The intersection of these two lines will give us the "P" we are looking for

 

Midpoint of OA  =  (5, - 5)

Equation of a line through this point perpendicular to y = -x  + 6  is

 

y = (x -5) - 5

y = x - 10

 

Find the x coordinate of the intersection of these lines

 

-x + 6  = x - 10

2x = 16

x = 8

And y = -8 + 6  =  -2

 

So P  =  (8, -2)

 

And we can see that this point makes PO = PA

 

 

 

cool cool cool

 Jun 11, 2019

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