In a tetrahedron OABC, the edges are of lengths, `|OA|=|BC|=a,|OB|=|AC|=b,|OC|=|AB|=c.` Let `G_1 and G_2` be the centroids of the triangle ABC and AOC such that `OG_1 _|_ BG_2,` then the value of `(a^2+c^2)/b^2` is

Options

A. 2
B. 3
C. 6
D. 9

Answer

To solve the problem, we need to find the value of \((a^2 + c^2)/b^2\) given the conditions of the tetrahedron \(OABC\) and the centroids \(G_1\) and \(G_2\).

### Step-by-Step Solution:

1. **Define the Vectors**:
Let the position vectors of points \(O\), \(A\), \(B\), and \(C\) be represented as:
\[ \vec{OA} = \vec{a}, \quad \vec{OB} = \vec{b}, \quad \vec{OC} = \vec{c} \]
Given that:
\[ |\vec{OA}| = |\vec{BC}| = a, \quad |\vec{OB}| = |\vec{AC}| = b, \quad |\vec{OC}| = |\vec{AB}| = c \]

2. **Find the Centroids**:
The centroid \(G_1\) of triangle \(ABC\) is given by:
\[ \vec{G_1} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \]
The centroid \(G_2\) of triangle \(AOC\) is given by:
\[ \vec{G_2} = \frac{\vec{a} + \vec{c}}{3} \]

3. **Condition of Perpendicularity**:
We are given that \(\vec{OG_1} \perp \vec{BG_2}\). This translates to:
\[ \vec{OG_1} \cdot \vec{BG_2} = 0 \]
Where:
\[ \vec{OG_1} = \vec{G_1} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \]
and
\[ \vec{BG_2} = \vec{G_2} - \vec{b} = \frac{\vec{a} + \vec{c}}{3} - \vec{b} = \frac{\vec{a} + \vec{c} - 3\vec{b}}{3} \]

4. **Dot Product Calculation**:
Now, substituting into the dot product condition:
\[ \left(\frac{\vec{a} + \vec{b} + \vec{c}}{3}\right) \cdot \left(\frac{\vec{a} + \vec{c} - 3\vec{b}}{3}\right) = 0 \]
This simplifies to:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{c} - 3\vec{b}) = 0 \]

5. **Expanding the Dot Product**:
Expanding the dot product gives:
\[ \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{c} - 3\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} + \vec{b} \cdot \vec{c} - 3\vec{b} \cdot \vec{b} + \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} - 3\vec{c} \cdot \vec{b} = 0 \]
Using the fact that \(\vec{a} \cdot \vec{b} = 0\), \(\vec{b} \cdot \vec{c} = 0\), and \(\vec{c} \cdot \vec{a} = 0\) (since the edges are mutually perpendicular), we simplify this to:
\[ |\vec{a}|^2 + |\vec{c}|^2 - 3|\vec{b}|^2 = 0 \]

6. **Final Equation**:
Rearranging gives:
\[ |\vec{a}|^2 + |\vec{c}|^2 = 3|\vec{b}|^2 \]
Therefore, we can express this as:
\[ \frac{|\vec{a}|^2 + |\vec{c}|^2}{|\vec{b}|^2} = 3 \]

### Conclusion:
Thus, the value of \(\frac{a^2 + c^2}{b^2}\) is \(3\).