Question

Factorize by the grouping method :

$(ax+by{)}^2+(bx-ay{)}^2$

• A. $(x^2+y^2)(a^{}+b^2)$
• B. $(x^2+y^2)(a^2+b^2)$
• C. $(x^2-y^2)(a^2-b^3)$
• D. $(x^3-y^2)(a^3+b^2)$

Answer 1

The correct option is B $(x^2+y^2)(a^2+b^2)$

$(ax+by{)}^2+(bx-ay{)}^2$
$=(ax{)}^2+2(ax\left)\right(by)+(by{)}^2+(bx{)}^2-2(bx\left)\right(ay)+(ay{)}^2$
$=a^2{x}^2+2abxy+b^2{y}^2+b^2{x}^2-2abxy+a^2{y}^2$
$=a^2{x}^2+b^2{x}^2+b^2{y}^2+a^2{y}^2$
$=x^2(a^2+b^2)+y^2(a^2+b^2)$
$=(a^2+b^2)(x^2+y^2)$