Question

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

• A. no solution
• B. unique solution
• C. infinitely many solutions
• D. finitely many solutions

Answer 1

The correct option is B unique solution

$\Delta =\left|\begin{array}{ccc}\frac{1}{a^2}& \frac{1}{b^2}& -\frac{1}{c^2}\\ \frac{1}{a^2}& -\frac{1}{b^2}& \frac{1}{c^2}\\ -\frac{1}{a^2}& \frac{1}{b^2}& \frac{1}{c^2}\end{array}\right|=\frac{1}{a^2{b}^2{c}^2}\left|\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right|$

$=\frac{1}{a^2{b}^2{c}^2}\left|\begin{array}{ccc}2& 0& 0\\ 1& -1& 1\\ -1& 1& 1\end{array}\right|=\frac{-4}{a^2{b}^2{c}^2}\ne 0$
$\therefore$ Given system has unique solution.
Hence, option B.