question_answer

If the circle\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] touches by the line \[y=x\] at the point \[P\] such that \[OP=6\sqrt{2}\], where \[O\] is the origin, then the value of \[c\] is equal to

A) \[74\]                                   

B) \[62\]

C) \[64\]                                   

D) \[72\]

Correct Answer: D

Solution :

The equation of the line y = x in parametric form is                 \[\frac{x}{\cos \pi /4}=\frac{y}{\sin \pi /4}\] \[\because \]     \[OP=6\sqrt{2}\] \[\therefore \]Coordinates of \[P\] are given by                 \[\frac{x}{\cos \frac{\pi }{4}}=\frac{y}{\sin \frac{\pi }{4}}=6\sqrt{2}\] \[\Rightarrow \]               \[x=y=6\] \[\therefore \]Coordinates of\[P\]are\[(6,\,\,6).\] The equation of circle touching\[y=x\]at\[P\,\,(6,\,\,6)\]is                 \[{{(x-6)}^{2}}+{{(y-6)}^{2}}+\lambda (x-y)=0\] \[\Rightarrow \]               \[{{x}^{2}}+{{y}^{2}}+x(\lambda -12)-y(\lambda +12)+72=0\] Comparing it with\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]                 \[\lambda -12=2g,\,\,-(\lambda +12=2f\]and\[c=72\] Hence, required value of \[c\] is\[72\].