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)
p2+q2+2p(−‌
h+p
2
)+2q(−‌
k+q
2
)+(‌
h+p
2
)2=0
⇒h2+p2−2hp−4kq=0 ∴ locus of (h,k) is x2+p2−2xp−4yq=0 ⇒(x−p)2=4qy