Question

$\text{If}X=1-a^2\text{and}Y=1-b^2,\text{then what is}{X}^2+Y^2?$

• A. $a^4+b^4-2(a^2+b^2)+2$
• B. $a^4+b^4-2(a^2-b^2)$
• C. $a^4+b^4+2(a^2-b^2)+2$
• D. $a^4+b^4+2(a^2-b^2)$

Answer 1

The correct option is A $a^4+b^4-2(a^2+b^2)+2$

$X=1-a^{2}Y=1-b^2{X}^2=(1-a^2{)}^2=1+(a^2{)}^2-2\times 1\times a^2=1+a^4-2a^2{Y}^2=(1-b^2{)}^2=1+(b^2{)}^2-2\times 1\times b^2=1+b^4-2b^2\therefore X^2+Y^2=1+a^4-2a^2+1+b^4-2b^2\Rightarrow X^2+Y^2=a^4+b^4-2a^2-2b^2+2\Rightarrow X^2+Y^2=a^4+b^4-2(a^2+b^2)+2$