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Let $a,b,c$ be positive real numbers. The following system of equations in $x, y$ and $ z $ $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1$, $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1, - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$ has
No solution
Unique solution
Infinitely many solutions
Finitely many solutions
Solution
(b) Let $\frac{{{x^2}}}{{{a^2}}} = X,\frac{{{y^2}}}{{{b^2}}} = Y$and $\frac{{{z^2}}}{{{c^2}}} = Z$,
then the given system of equations is $X + Y – Z = 1,$ $X – Y + Z = 1$, $ – X + Y + Z = 1$.
The coefficient matrix is $A = \left[ {\begin{array}{*{20}{c}}1&1&{ – 1}\\1&{ – 1}&1\\{ – 1}&1&1\end{array}} \right]$
Clearly $|A| \ne 0$. So the given system of equations has unique solution.
