2025 · Q38
If x^4 + y^4 = 14x^2y^2, consider:
- Ilog10(x^2+y^2)=log10x+log10y+2log10 2
- IIlog10(x^2-y^2)=log10x+log10y+log10 2+0.5log10 3. Which is/are correct?
Elementary MathematicsAlgebra
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
(x^2+y^2)^2=x^4+2x^2y^2+y^4=16x^2y^2, so x^2+y^2=4xy and log10(x^2+y^2)=log10(4xy)=log10(x)+log10(y)+2log10(2) (I true). (x^2-y^2)^2=12x^2y^2 gives |x^2-y^2|=2*sqrt(3)*xy, so log10(x^2-y^2)=log10(x)+log10(y)+log10(2)+0.5log10(3) (II true). Both hold. The trick is completing the square to 16x^2y^2 and 12x^2y^2.