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Question

If one end of a diameter of a circle is x2+y28x12y+22=0 is (2,3) then find the co-ordinates of the other end of the diameter.

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Solution

The given circle is x2+y28x12y+22=0 .......(1)

is of the form x2+y2+2gx+2fy+c=0

Its centre is C(g,f)=C(4,6)

Let A(2,3) and B(α,β) be the ends of the diameter, then C(4,6) is the midpoint of the segment AB

2+α2=4 and 3+β2=6

2+α=8 and 3+β=12

α=82=6 and β=123=9

Hence, the coordinates of the other end of the diameter are (6,9)

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