Let a, b, c be positive numbers, the following systems of equations in x, y and z \( \Large \frac{x^{2}}{a^{2}} +\ \frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1; \ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} +\ \frac{z^{2}}{c^{2}}=1 and\ \frac{-x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \) has;

Given system of equations are

\( \Large \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{zc}{c^{2}} = 1, \)

\( \Large \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1 \)

and \( \Large -\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1 \)

On adding all these equations, we get,

 \( \Large \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 3 \)

\( \Large \frac{2z^{2}}{c^{2}}=2,\ \frac{2y^{2}}{b^{2}}=2,\ \frac{ax^{2}}{a^{2}} = 2 \)

On subtracting (i) from (iv), (ii) equation from (iv) and (iii) from (iv) we get

=> \( \Large z = \pm \ c,\ y = \pm \ b,\ x = \pm \ a \)