Question

Add: ${\mathrm{x}}^2-{\mathrm{y}}^2-1,{\mathrm{y}}^2-1-{\mathrm{x}}^2,1-{\mathrm{x}}^2-{\mathrm{y}}^2$

Answer 1

We have,

${\mathrm{x}}^2-{\mathrm{y}}^2-1,{\mathrm{y}}^2-1-{\mathrm{x}}^2,1-{\mathrm{x}}^2-{\mathrm{y}}^2$

On combining the like terms and adding them, we get

$$\begin{gathered} ={\mathrm{x}}^2–{\mathrm{y}}^2–1+({\mathrm{y}}^2–1–{\mathrm{x}}^2)+(1–{\mathrm{x}}^2–{\mathrm{y}}^2) \\ ={\mathrm{x}}^2–{\mathrm{y}}^2–1+{\mathrm{y}}^2–1–{\mathrm{x}}^2+1–{\mathrm{x}}^2–{\mathrm{y}}^2 \\ ={\mathrm{x}}^2–{\mathrm{x}}^2–{\mathrm{x}}^2–{\mathrm{y}}^2+{\mathrm{y}}^2–{\mathrm{y}}^2–1–1+1 \\ ={\mathrm{x}}^2(1–1-1)+{\mathrm{y}}^2(-1+1–1)+(-1-1+1) \\ ={\mathrm{x}}^2(1–2)+{\mathrm{y}}^2(-2+1)+(-2+1) \\ ={\mathrm{x}}^2(-1)+{\mathrm{y}}^2(-1)+(-1) \\ =-{\mathrm{x}}^2–{\mathrm{y}}^2-1 \end{gathered}$$

Hence, when we add ${\mathrm{x}}^2-{\mathrm{y}}^2-1,{\mathrm{y}}^2-1-{\mathrm{x}}^2,1-{\mathrm{x}}^2-{\mathrm{y}}^2$ we get $-{\mathrm{x}}^2–{\mathrm{y}}^2-1$.