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for what values of m is the line y=mx+2 at a distance 4 from (1,3)

Mathematics
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@Whatever1, Welcome to OpenStudy. I hope you enjoy your stay here. If you have any questions about the site ask the ambassadors. They have a purple Ⓐ by there name :) Or you could always shoot me a PM. @ParthKohli will now answer your question ^_^.
Okay, I thought OP would do it, but I will. First, we have four points that are at distance 4 from \((1,3)\):\[(1,7) \\ (1,-1) \\ (5,3) \\ (-3,3)\]There is another point but we'd look at it later.
Now, you just have to figure out the slopes where \(y = mx + 2\) passes through these points. Can you do that?

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There are actually many more points. There are infinite such points in fact. O_O
@Opcode shall help you with the rest. :-)

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