find the equation of a circle through (0,5),(3,4), touching the line y+5=0.

The general equation of a circle has three variables, if
the center is (a, b) and r is the radius: (x - a)^2 + (y - b)^2 =
r^2.

You have provided two points through which the circle
passes (0,5) and (3,4) and a tangent line to the circle y + 5 =
0

Now, as the line y + 5 = 0 is a tangent, the y-coordinate
of the point that touches the line is -5. The point (0, 5) lies on the other end of the
circle. This gives the radius of the circle as 5.

So we
have a^2 + (5 - b)^2 = 25

and (3 - a)^2 + (4 - b)^2 =
25

(3 - a)^2 - a^2 + (4 - b)^2 - (5 - b)^2 =
25

=> (3 - a - a)(3 - a + a) + (4 - b - 5 + b)(4 - b
+ 5 - b) = 0

=> 3(3 - 2a) - 1(9 - 2b) =
0

=> 9 - 6a - 9 + 2b =
0

=> 6a = 2b

=>
3a = b

Substitute in a^2 + (5 - b)^2 =
25

=> a^2 + (5 - 3a)^2 =
25

=> a^2 + 25 + 9a^2 - 30a =
25

=> 10a^2 - 30a =
0

=> 10a(a - 3) =
0

=> a = 0 and a = 3

b
= 0 and b = 9

The equation of the required
circle can be x^2 + y^2 = 25 and (x - 3)^2 + (y - 9)^2 =
25