2026 · Q3
If a^2/(b^2 + c^2) = b^2/(c^2 + a^2) = c^2/(a^2 + b^2), then what is the value of a^4 + b^4 + c^4 equal to?
Elementary MathematicsAlgebra
- Aa^2b^2 + b^2c^2 + c^2a^2
- B2(a^2b^2 + b^2c^2 + c^2a^2)
- C3(a^2b^2 + b^2c^2 + c^2a^2)
- D4(a^2b^2 + b^2c^2 + c^2a^2)
Explanation
Let the common value of the three ratios be t, so a²=t(b²+c²), b²=t(c²+a²), c²=t(a²+b²). Adding the three equations gives a²+b²+c²=2t(a²+b²+c²), so t=1/2 and hence 2a²=b²+c² (and cyclically). Multiplying each relation by the corresponding square, a⁴=a²(b²+c²)/2, and summing the three gives a⁴+b⁴+c⁴=a²b²+b²c²+c²a², option A.