JEE Main & Advanced Mathematics Circle and System of Circles Equation of a Circle in Some Special Cases

(1) If centre of the circle is \[(h,\,k)\] and it passes through origin then its equation is \[{{(x-h)}^{2}}+{{(y-k)}^{2}}={{h}^{2}}+{{k}^{2}}\]\[\Rightarrow \,\,{{x}^{2}}+{{y}^{2}}\] \[-2hx-2ky=0\].

(2) If the circle touches x-axis then its equation is  \[{{(x\pm h)}^{2}}+{{(y\pm k)}^{2}}={{k}^{2}}\]. (Four cases)

 (3) If the circle touches y-axis then its equation is  \[{{(x\pm h)}^{2}}+{{(y\pm k)}^{2}}={{h}^{2}}\]. (Four cases)    

(4) If the circle touches both the axes then its equation is \[{{(x\pm r)}^{2}}+{{(y\pm r)}^{2}}={{r}^{2}}\] . (Four cases)    

(5) If the circle touches x- axis at origin then its equation is  \[{{x}^{2}}+{{(y\pm k)}^{2}}={{k}^{2}}\] \[\Rightarrow \,\,{{x}^{2}}+{{y}^{2}}\pm 2ky=0\]. (Two cases)    

(6) If the circle touches y-axis at origin, the equation of circle is \[{{(x\pm h)}^{2}}+{{y}^{2}}={{h}^{2}}\]\[\Rightarrow \,\,{{x}^{2}}+{{y}^{2}}\pm 2xh=0\]. (Two cases)    

(7) If the circle passes through origin and cut intercepts \[a\] and \[b\] on axes, the equation of circle is \[{{x}^{2}}+{{y}^{2}}-ax-by=0\] and centre is \[C(a/2,\,\,b/2)\]. (Four cases)