Quadratic Equations
Algebraic Inequalities
JEE Advanced
Grade 11
Question:
If $a^2 + b^2 + c^2 = 1$, then $ab + bc + ca$ lies in:
(1) $[-\frac{1}{2}, \frac{1}{2}]$
(2) $[1, 2]$
(3) $[-\frac{1}{2}, 1]$
(4) $[-1, 2]$
Step-by-Step Solution
Key Concept: For the lower bound use $(a+b+c)^2 \geq 0$; for the upper bound use $(a-b)^2+(b-c)^2+(c-a)^2 \geq 0$. Both give inequalities on $ab + bc + ca$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)