The circle passes through the origin and makes intercepts a with x-axis and b with y-axis.
                          OA = a, OB = b
∴ The co-ordinates of A are (a, 0) and of B are (0, b)
Now, we have to find the equation of a circle which passes through the points O (0, 0), A (a, 0), B (0, b).
                                
Let the equation of the circle
                                                ...(i)
Since it passes through (0, 0)
∴                 
                                                           ...(ii)
Since it passes through A (a, 0)
∴        
                                          ...(iii)

But, from (ii),  
∴ from (iii), we have 
Since the circle passes through (0, b)
∴    
                       
∴ from (ii), we have 
∴ The equation of circle is: 
or    
Hence, the equation of circle is : 

Centre (h, k)   
∴ Equation of circle is : 
or 
or  
or  
or  
Hence, the equation of circle is: 

Centre (h, k)  (-2, 3)
                       h = -2, k = 3, Radius, r = 4
∴ The equation of circle is : 
or      

Centre (h, k) (-a, -b)  h = -a, k = -b
     Diameter of circle = 
∴ Equation of circle is : 
or     
    
Hence, the equation of circle is: 

The centre of circle is C (h, k)  (2, 2)  h = 2, k =2
Since the circle passes through the point A (4, 5).
∴ Radius of the circle is : r = CA
  
∴ Equation of circle is: 
or   
Hence, the equation of circle is