question_answer

            Let a, b and c denote the sides BC, CA and AB Respectively of \[\Delta ABC.\]If \[\left| \begin{matrix}    1 & a & b  \\    1 & c & a  \\    1 & b & c  \\ \end{matrix} \right|=0,\] then find the value of             \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C.\]

Answer:

We have,             \[\Rightarrow \]                 \[[applying\,\,{{R}_{2}}\to {{R}_{2}}-{{R}_{1}},\,\,{{R}_{3}}\to {{R}_{3}}-{{R}_{1}}]\] \[\Rightarrow \]   [expanding along \[{{C}_{1}}\]] \[\Rightarrow \]   \[(c-a)(c-b)-(a-b)(b-a)=0\] \[\Rightarrow \]   \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca=0\] \[\Rightarrow \]   \[2{{a}^{2}}+2{{b}^{2}}+2{{c}^{2}}-2ab-2bc-2ca=0\] \[\Rightarrow \]   \[{{(a-b)}^{2}}+{{(b-c)}^{2}}+{{(c-a)}^{2}}=0\] \[\Rightarrow \]   \[a-b=0,\] \[b-c=0,\] \[c-a=0\] \[\Rightarrow \]   a = b = c \[\Rightarrow \] \[\Delta ABC\] is an equilateral triangle. \[\Rightarrow \]   \[A=B=C=\frac{\pi }{3}\] \[\therefore \]      \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C=3{{\sin }^{2}}\frac{\pi }{3}\] \[=3\times {{\left( \frac{\sqrt{3}}{2} \right)}^{2}}\]                         \[=\frac{9}{4}\]