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Question

A variable circle passes through the point P(1,2) and touches the xaxis. The locus of the other end of the diameter through P is
  1. (x1)2=8y
  2. (y1)2=8x
  3. (x1)2+8y=0
  4. (x+1)2=8y


Solution

The correct option is A (x1)2=8y
Equation of any circle touching xaxis is of the form
(xh)2+(yk)2=k2     (1)
Let the coordinates of the other end of the diameter through P be (α,β)
Then, α+12=h and β+22=k

Also, putting P(1,2) in (1), we get
(1h)2+(2k)2=k2
or (h1)2+(k2)2=k2
(α+121)2+(β+222)2=(β+22)2
(α1)2+(β2)2=(β+2)2
(α1)2=8β
Locus of (α,β) is (x1)2=8y

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