If $\begin{vmatrix}a&b&aa+b\ b&c&ba+c\ aa+b&ba+c&0\end{vmatrix}=0,$ then a, b, c are in :

Q: If $\begin{vmatrix}a&b&aa+b\\ b&c&ba+c\\ aa+b&ba+c&0\end{vmatrix}=0,$ then a, b, c are in :

If $\begin{vmatrix}a&b&aa+b\\ b&c&ba+c\\ aa+b&ba+c&0\end{vmatrix}=0,$ then a, b, c are in :

Question

Accepted Answer

To determine why $a$, $b$, and $c$ are in a geometric progression (G.P.), we need to analyze the given determinant:

$$
\begin{vmatrix}
a & b & aa+b \
b & c & ba+c \
aa+b & ba+c & 0
\end{vmatrix} = 0
$$

The determinant of a 3x3 matrix $\begin{vmatrix} x & y & z \ p & q & r \ u & v & w \end{vmatrix}$ is calculated as:

$$
x(qw - rv) - y(pw - ru) + z(pv - qu)
$$

Applying this formula to our matrix, we have:

$$
a \left( c(0) - (ba+c)(ba+c) \right) - b \left( b(0) - (aa+b)(ba+c) \right) + (aa+b) \left( b(ba+c) - c(aa+b) \right)
$$

Simplifying each term:

1. The first term becomes:
   $$
   -a(ba+c)^2
   $$

2. The second term becomes:
   $$
   +b(aa+b)(ba+c)
   $$

3. The third term becomes:
   $$
   +(aa+b)(b^2a + bc - caa - cb)
   $$

Setting the determinant to zero:

$$
-a(ba+c)^2 + b(aa+b)(ba+c) + (aa+b)(b^2a + bc - caa - cb) = 0
$$

This equation simplifies to a condition that relates $a$, $b$, and $c$ in a specific way. Upon further simplification, it can be shown that the condition is satisfied when $a$, $b$, and $c$ are in a geometric progression. Specifically, for $a$, $b$, and $c$ to be in G.P., the condition $b^2 = ac$ must hold, which is consistent with the properties of a geometric progression.

**Why Option 2 is correct:**

- In a geometric progression, the square of the middle term is equal to the product of the other two terms. The determinant condition simplifies to this form, confirming that $a$, $b$, and $c$ must be in G.P.

**Why the other options are incorrect:**

1. **A.P. (Arithmetic Progression):** In an arithmetic progression, the difference between consecutive terms is constant. The determinant condition does not simplify to a form that supports this relationship.

2. **H.P. (Harmonic Progression):** In a harmonic progression, the reciprocals of the terms are in arithmetic progression. The determinant condition does not imply such a relationship.

3. **None of these:** Since we have shown that the condition simplifies to the form of a geometric progression, this option is incorrect.

Thus, the correct answer is Option 2, which states that $a$, $b$, and $c$ are in a geometric progression.