Question

$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1,\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1,-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ , has

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

Answer 1

Answer: (B)

unique solution

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 reduces to

$X+Y-Z=1,X-Y+Z=1$ and $-X+Y+Z=1$

The coefficient matrix $A$ of the above system of equations is given by

$\left[\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right]$

Now,$\left|A\right|=\left|\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right|$

$=1(-1-1)-1(1+1)-1(1-1)$

$=-2-2-0=-4$

Thus,$\left|A\right|\ne 0$

So, the given system of equations has a unique solution.