question_answer

The image of the centre of the circle\[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]with respect to the mirror \[x+y=1\]is

A)  \[\left( \frac{1}{\sqrt{2}},\sqrt{2} \right)\]                          

B)  \[(\sqrt{2},\sqrt{2})\]

C)  \[(\sqrt{2},2\sqrt{2})\]

D)         \[(-\sqrt{2},2)\]

E)  None of these

Correct Answer: E

Solution :

Let\[p\]be image of the origin in the line \[x+y=1\]. Since,\[OA=OB,\]therefore Q is the midpoint of AB. \[\therefore \]coordinates of Q are\[\left( \frac{1}{2},\frac{1}{2} \right)\]. Let the coordinates of P be\[({{x}_{1}},{{y}_{1}})\]. Since, Q is the midpoint of OP. \[\therefore \] \[\frac{0+{{x}_{1}}}{2}=\frac{1}{2}\]and \[\frac{0+{{y}_{1}}}{2}=\frac{1}{2}\] \[\Rightarrow \]               \[{{x}_{1}}=1,{{y}_{1}}=1\] \[\therefore \]The coordinates of \[P\]are (1, 1).