Equation of circles touching x-axis at the origin and the line 4x–3y+24=0 are 
 

$$\begin{gathered} Let the equation of circle pas\sin g through (0,0) is \\ {x}^2+y^2+2gx+2fy=0 then \\ {g}^2=c\Rightarrow g=0 \\ {x}^2+y^2+2fy=0 \\ centre=(0,-f) and r=f \\ it touches the line 4x-3y+24=0 then \\ r=d \\ \Rightarrow f=\frac{\left|4\left(0\right)-3(-f)+24\right|}{\sqrt{25}} \\ \Rightarrow 5f=\left|3f+24\right| \\ \Rightarrow f=-3 or 12 \\ Required circle is \\ {x}^2+y^2-6y=0 or x^2+y^2+24y=0 \end{gathered}$$