Question

Let $a,b,c>R^{+}$ ( i.e. $a,b,c$ are positive real numbers) then the following system of equations in $x,y,z$
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$, $\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ and $\frac{-x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ has

• A. No solution
• B. Unique solution
• C. Infinitely many solution
• D. Finitely many solution

Answer 1

The correct option is B Unique solution

Put $\frac{x^2}{a^2}=X,\frac{y^2}{b^2}=Y,\frac{z^2}{c^2}=Z$
So the given equations becomes
$X+Y-Z=1$
$X-Y+Z=1$
$-X+Y+Z=1$
So coefficient matrix A $=\left[\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right]$
Now, $\left|A\right|=-4\ne 0$, so system of equations has a unique solution.