Find the equation of circle touches x axis at 3,0 and through the point 1,2

Let the eq of circle be

x
2
+y
2
+2gx+2βy+c=0

now, x intercept = 2
g
2
−c


But circle touches x axis ∴ x intercept = 0

∴2
g
2
−c

=0∴g
2
=c

∴eq
n
become x
2
+y
2
+3gx+2βy+g
2
=0

Now circle passes through (3, 0) and (1, 4)

∴(3)
2
+(0)
2
+2g(3)+0+g
2
=0

9+6g+g
2
=0

∴g=−3

Also (1)
2
+(4)
2
+2(−3)(1)+2β(4)+9=0

∴1+16−6+8β+9=0

∴20+8β=0 ∴β=
2
5


Hence eq
n
of circle is

x
2
+y
2
−6x−5y+9=