If \(x=1-y\) and \(x^2=2-y^2\), then find the value of \((xy)^2+5\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{21}{4}\)
The two conditions give the sum \(x+y\) and the sum of squares \(x^2+y^2\). The product comes from \((x+y)^2=x^2+y^2+2xy\).
\(x+y=1\) and \(x^2+y^2=2\)
\(1^2=2+2xy\), so \(2xy=-1\) and \(xy=-\displaystyle\frac{1}{2}\)
\((xy)^2=\displaystyle\frac{1}{4}\)
\((xy)^2+5=\displaystyle\frac{1}{4}+5=\frac{21}{4}\)