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Question

If (2,0), (0,1), (4,5) and (0,c) are concyclic, then find c.

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Solution

Let the center of circle passing through the points be (h,k)
The distance from center to all four points are equal
We have (h2)2+(k0)2=(h0)2+(k1)2
4h2k=3
We have (h2)2+(k0)2=(h4)2+(k5)2
4h+10k=37
By solving those two equations , we get h=176 and k=256
The radius of circle is 2536+62536=65036=5626
We have (1760)2+(256c)2=65036
(256c)2=36136
256c=196,196
c=1,223
Therefore the value of c is 223

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