Question
If $${a^2} + {b^2} + {c^2} = 1,\,\,{\text{then }}\,ab + bc + ca$$ lies in the interval
A.
$$\left[ {\frac{1}{2},2} \right]$$
B.
$$\left[ { - 1,2} \right]$$
C.
$$\left[ { - \frac{1}{2},1} \right]$$
D.
$$\left[ { - 1,\frac{1}{2}} \right]$$
Answer :
$$\left[ { - \frac{1}{2},1} \right]$$
Solution :
$$\eqalign{
& {\text{Given that }}\,{a^2} + {b^2} + {c^2} = 1\,\,\,\,\,\,\,\,\,\,\,.....\left( 1 \right) \cr
& {\text{We know }}{\left( {a + b + c} \right)^2} \geqslant 0 \cr
& \Rightarrow \,{a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca \geqslant 0 \cr
& \Rightarrow \,2\left( {ab + bc + ca} \right) \geqslant - 1\,\,\,\,\,\,\,\,\,\left[ {{\text{Using}}\left( 1 \right)} \right] \cr
& \Rightarrow \,ab + bc + ca \geqslant - \frac{1}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.....\left( 2 \right) \cr
& {\text{Also we know that}} \cr
& \frac{1}{2}\left[ {{{\left( {a - b} \right)}^2} + {{\left( {b - c} \right)}^2} + {{\left( {c - a} \right)}^2}} \right] \geqslant 0 \cr
& \Rightarrow \,{a^2} + {b^2} + {c^2} - ab - bc - ca \geqslant 0 \cr
& \Rightarrow \,ab + bc + ca \leqslant 1\,\,\,\,\,\,\,\,\,\left[ {{\text{Using}}\left( 1 \right)} \right]\,\,\,\,\,.....\left( 3 \right) \cr
& \,\,\,\,\,{\text{Combining}}\left( 2 \right){\text{and}}\left( 3 \right){\text{,we get}} \cr
& \,\, - \frac{1}{2} \leqslant ab + bc + ca \leqslant 1 \cr
& \therefore \,\,ab + bc + ca \in \left[ { - \frac{1}{2},1} \right] \cr
& \therefore \,\,\left( {\text{C}} \right){\text{is the correct answer}}{\text{.}} \cr} $$