question_answer

The coordinates of the centre of a circle passing through \[(1,2),(3,-4)\]and \[(5,-6)\] is _____.

A)  \[(2,11)\]                    

B)  \[(11,2)\]        

C)         \[(11,-2)\]      

D)         \[(-2,11)\]       

Correct Answer: B

Solution :

Let \[P(x,y)\] be the centre of the circle passing through \[A(1,2),\,B(3,-4)\] and \[C(5,-6)\] Also, we know that,                         \[AP=PB=PC\] Now, we have, \[A{{P}^{2}}=P{{B}^{2}}\] \[\Rightarrow \]     \[{{(x-1)}^{2}}+{{(y-2)}^{2}}\]             \[={{(x-3)}^{2}}+{{(y-2)}^{2}}\] \[\Rightarrow \]            \[x-3y=5\]                    ?..(i) Also,   \[A{{P}^{2}}=C{{P}^{2}}\] \[\Rightarrow \]            \[{{(x-1)}^{2}}+{{(y-2)}^{2}}\]             \[={{(x-5)}^{2}}+{{(y+6)}^{2}}\] \[\Rightarrow \]            \[x-2y=7\] Solving (i) and (ii), we get, \[x=22,\,\,\,y=2\] \[\therefore \]  The coordinates of centre \[P=(11,2)\]