A) \[\left( \frac{1}{2},-4 \right)\]
B) \[\left( -\frac{1}{2},4 \right)\]
C) \[\left( \frac{1}{2},4 \right)\]
D) \[\left( -\frac{1}{2},-4 \right)\]
Correct Answer: C
Solution :
Let the coordinates of the required point be \[({{x}_{1}},{{y}_{1}}),\] which is equidistant from the points \[(0,0),\] \[(10,8)\] and \[(4,6)\]. \[\therefore \] \[{{({{x}_{1}}-0)}^{2}}+{{({{y}_{1}}-0)}^{2}}={{({{x}_{1}}-0)}^{2}}+{{({{y}_{1}}-8)}^{2}}\] \[={{({{x}_{1}}-4)}^{2}}+{{({{y}_{1}}-6)}^{2}}\] or \[x_{1}^{2}+y_{1}^{2}=x_{1}^{2}+y_{1}^{2}-16y+64\] \[=x_{1}^{2}+y_{1}^{2}-8x-12y+16+36\] or \[16y=64\] or \[y=4\] and \[2x+3y=13\] or \[x=\frac{1}{2}\] \[\therefore \] Coordinates of the required point will be \[\left( \frac{1}{2},4 \right)\].
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