已知实数a满足a2+ab+b2=3,a2-ab+b2=k,求k的取值范围.

题目

已知实数a满足a2+ab+b2=3,a2-ab+b2=k,求k的取值范围.

答案

a^2+ab+b^2=3
==>
a^2+b^2=3-ab≥2ab
==>
ab≤1
a^2+b^2=3-ab≥2|ab|≥-2ab
==>
ab≥-3
==>
-3≤ab≤1
==>
1≤3-2ab≤9
∴k=a^2-ab+b^2=3-2ab
范围是【1,9】