question_answer

The equation of a circle touching the axes of coordinates and the line \[x\cos \alpha +y\sin \alpha =2\]can be

A)            \[{{x}^{2}}+{{y}^{2}}-2gx-2gy+{{g}^{2}}=0\], where \[g=\frac{2}{(\cos \alpha +\sin \alpha +1)}\]

B)            \[{{x}^{2}}+{{y}^{2}}-2gx-2gy+{{g}^{2}}=0\], where \[g=\frac{2}{(\cos \alpha +\sin \alpha -1)}\]

C)            \[{{x}^{2}}+{{y}^{2}}-2gx+2gy+{{g}^{2}}=0\], where \[g=\frac{2}{(\cos \alpha -\sin \alpha +1)}\]                          

D)            \[{{x}^{2}}+{{y}^{2}}-2gx+2gy+{{g}^{2}}=0\] where \[g=\frac{2}{(\cos \alpha +\sin \alpha +1)}\]

E)            All of these

Correct Answer: E

Solution :

           \[{{x}^{2}}+{{y}^{2}}-2gx-2gy+{{g}^{2}}=0\]            \[g=\pm \frac{g\cos \alpha +g\sin \alpha -2}{\sqrt{{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha }}\] Þ \[g=\frac{2}{\sin \alpha +\cos \alpha \pm 1}\].                    Similarly other options hold.