SOLUTION: Find the coordinates of all points that are on the line x=y and are five units from (3,-1).

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Question 517509: Find the coordinates of all points that are on the line x=y and are five units from (3,-1).
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find the coordinates of all points that are on the line x=y and are five units from (3,-1).
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It's the 2 points of intersection of the line and a circle of radius 5 centered at (3,-1).
The circle is %28x-3%29%5E2+%2B+%28y%2B1%29%5E2+=+25
x = y
%28x-3%29%5E2+%2B+%28x%2B1%29%5E2+=+25
2x%5E2+-+4x+-+15+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-4x%2B-15+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-4%29%5E2-4%2A2%2A-15=136.

Discriminant d=136 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+136+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+136+%29%29%2F2%5C2+=+3.91547594742265
x%5B2%5D+=+%28-%28-4%29-sqrt%28+136+%29%29%2F2%5C2+=+-1.91547594742265

Quadratic expression 2x%5E2%2B-4x%2B-15 can be factored:
2x%5E2%2B-4x%2B-15+=+%28x-3.91547594742265%29%2A%28x--1.91547594742265%29
Again, the answer is: 3.91547594742265, -1.91547594742265. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-4%2Ax%2B-15+%29

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x,y = 1 + sqrt(8.5)
x,y = 1 - sqrt(8.5)