Circles Part 1

Let the variable circle be
x2+y2+2gx+2fy+c=0........(1)
∴p2+q2+2gp+2fq+c=0......(2)
Circle (1) touches x-axis,
∴g2−c=0⇒c=g2.
From (2)
p2+q2+2gp+2fq+g2=0.....(3)
Let the other end of diameter through (p,q) be (h,k),
then, ‌

h+p

2

=−g and ‌

k+q

2

=−f
Put in (3)
⇒h2+p2−2hp−4kq=0
∴ locus of (h,k) is x2+p2−2xp−4yq=0
⇒(x−p)2=4qy