From the sum of $x^{2} - y^{2} - 1, y^{2} - x^{2} - 1$ and $1 - x^{2} - y^{2}$ , subtract $- (1 + y^{2}) .$

- x^{2}

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- y^{2}

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Solution

The correct option is A $- x^{2}$
Sum of $x^{2} - y^{2} - 1, y^{2} - x^{2} - 1$ and $1 - x^{2} - y^{2}$
$= x^{2} - y^{2} - 1 + y^{2} - x^{2} - 1 + 1 - x^{2} - y^{2}$
On grouping the like terms.
$= (x^{2} - x^{2} - x^{2}) + (- y^{2} + y^{2} - y^{2}) + (- 1 - 1 + 1)$
On solving the groups,
$= - x^{2} - y^{2} - 1$
Now, subtract $- (1 + y^{2})$ from $- x^{2} - y^{2} - 1$
$= - x^{2} - y^{2} - 1 - [- (1 + y^{2})] = - x^{2} - y^{2} - 1 + (1 + y^{2}) = - x^{2} - y^{2} - 1 + 1 + y^{2} = - x^{2} - y^{2} + y^{2} - 1 + 1 = - x^{2}$

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