A) \[\sqrt{(a_{1}^{2}+b_{1}^{2}+c_{1}^{2})}\]
B) \[a_{1}^{2}-b_{1}^{2}-c_{1}^{2}\]
C) \[\frac{1}{2}(a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2})\]
D) None of these
Correct Answer: C
Solution :
Let \[P(x,y)\] be any point on the line \[({{a}_{1}}-{{a}_{2}})x+({{b}_{1}}-{{b}_{2}})y+c=0\] ?(i) Since, \[{{(x-{{a}_{1}})}^{2}}+{{(y-{{b}_{1}})}^{2}}={{(x-{{a}_{2}})}^{2}}+{{(y-{{b}_{2}})}^{2}}\] \[\Rightarrow \] \[{{x}^{2}}+a_{1}^{2}-2{{a}_{1}}x+{{y}^{2}}+b_{1}^{2}-2{{b}_{1}}y\] \[={{x}^{2}}+a_{2}^{2}-2{{a}_{2}}x+{{y}^{2}}+b_{2}^{2}-2{{b}_{2}}y\] \[\Rightarrow \] \[2({{a}_{2}}-{{a}_{1}})x+2({{b}_{2}}-{{b}_{1}})y\] \[=a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2}\] \[\Rightarrow \] \[({{a}_{1}}-{{a}_{2}})x+({{b}_{1}}-{{b}_{2}})y\] \[+\frac{(a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2})}{2}=0\] ??(ii) Since, Eqs. (i) and (ii) represents the same equation of line. \[\therefore \] \[c=\frac{a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2}}{2}\]
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