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

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Solution

## 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$={\mathrm{x}}^{2}–{\mathrm{y}}^{2}–1+\left({\mathrm{y}}^{2}–1–{\mathrm{x}}^{2}\right)+\left(1–{\mathrm{x}}^{2}–{\mathrm{y}}^{2}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{x}}^{2}–{\mathrm{y}}^{2}–1+{\mathrm{y}}^{2}–1–{\mathrm{x}}^{2}+1–{\mathrm{x}}^{2}–{\mathrm{y}}^{2}\phantom{\rule{0ex}{0ex}}={\mathrm{x}}^{2}–{\mathrm{x}}^{2}–{\mathrm{x}}^{2}–{\mathrm{y}}^{2}+{\mathrm{y}}^{2}–{\mathrm{y}}^{2}–1–1+1\phantom{\rule{0ex}{0ex}}={\mathrm{x}}^{2}\left(1–1-1\right)+{\mathrm{y}}^{2}\left(-1+1–1\right)+\left(-1-1+1\right)\phantom{\rule{0ex}{0ex}}={\mathrm{x}}^{2}\left(1–2\right)+{\mathrm{y}}^{2}\left(-2+1\right)+\left(-2+1\right)\phantom{\rule{0ex}{0ex}}={\mathrm{x}}^{2}\left(-1\right)+{\mathrm{y}}^{2}\left(-1\right)+\left(-1\right)\phantom{\rule{0ex}{0ex}}=-{\mathrm{x}}^{2}–{\mathrm{y}}^{2}-1$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 $-{\mathrm{x}}^{2}–{\mathrm{y}}^{2}-1$.

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