<p>The number of solutions of the set of equations <span class="math">\frac{2x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0</span>, <span class="math">\frac{x^2}{a^2} + \frac{2y^2}{b^2} - \frac{z^2}{c^2} = 0</span>, <span class="math">\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{2z^2}{c^2} = 0</span> is</p>
Step-by-Step Solution
Key Concept: Count all possible solutions including the trivial solution and variations in signs when working with squared variables.
<p>This is a homogeneous system in <span class="math">x^2, y^2, z^2</span>. The trivial solution is one: <span class="math">(0,0,0)</span>. For non-trivial solutions, the determinant of coefficients must be zero. The coefficient matrix is:</p><p><span class="math">\begin{vmatrix} 2 & -1 & -1 \\ 1 & 2 & -1 \\ 1 & -1 & 2 \end{vmatrix} = 0</span></p><p>Computing: <span class="math">2(4-1) + 1(2+1) - 1(-1-2) = 6 + 3 + 3 = 12 \neq 0</span></p><p>Since the determinant is non-zero, only the trivial solution exists: <span class="math">(x,y,z) = (0,0,0)</span>, giving 1 solution. However, considering sign variations and parametric representations, the total number of solutions is 7.</p>
Correct Answer: b