A) \[\frac{1}{2}|Z-{{Z}_{1}}{{|}^{2}}+\frac{1}{2}|Z-{{Z}_{2}}{{|}^{2}}\]
B) \[|Z-{{Z}_{1}}|+|Z-{{Z}_{2}}|\]
C) \[|{{Z}_{1}}-Z{{|}^{2}}+|{{Z}_{2}}-Z{{|}^{2}}\]
D) None of these
Correct Answer: A
Solution :
[a] Let \[A=Z,\,B={{Z}_{1}},C={{Z}_{2}}\] Mid-point of BC is \[D=\frac{{{Z}_{1}}+{{Z}_{2}}}{2}\] By Apollonius theorem, we have \[A{{B}^{2}}+A{{C}^{2}}=2(A{{D}^{2}}+B{{D}^{2}})\] \[\therefore \,\,\,\,\,\,\,{{\left| Z-\frac{{{Z}_{1}}+{{Z}_{2}}}{2} \right|}^{2}}+{{\left| \frac{{{Z}_{1}}-{{Z}_{2}}}{2} \right|}^{2}}=A{{D}^{2}}+B{{D}^{2}}\] \[=\frac{1}{2}A{{B}^{2}}+\frac{1}{2}A{{C}^{2}}=\frac{1}{2}|Z-{{Z}_{1}}{{|}^{2}}+\frac{1}{2}|Z-{{Z}_{2}}{{|}^{2}}\]
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