If the three distinct lines
` x + 2ay + a = 0 , x+ 3by + b = 0` and
`x + 4ay + a = 0 ` are concurrent , then the point (a,b) lies on a .

Options

A. circle
B. straight line
C. parabola
D. hyperbola

Answer

To solve the problem, we need to determine the condition under which the three lines are concurrent. The lines given are:

1. \( x + 2ay + a = 0 \)
2. \( x + 3by + b = 0 \)
3. \( x + 4ay + a = 0 \)

### Step 1: Set up the determinant for concurrency
The lines are concurrent if the area of the triangle formed by their intersection points is zero. This can be determined using the determinant of the coefficients of the lines:

\[ \begin{vmatrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4a & a \end{vmatrix} = 0 \]

### Step 2: Calculate the determinant
We can calculate the determinant using the formula for a 3x3 matrix:

\[ D = 1 \cdot \begin{vmatrix} 3b & b \\ 4a & a \end{vmatrix} - 2a \cdot \begin{vmatrix} 1 & b \\ 1 & a \end{vmatrix} + a \cdot \begin{vmatrix} 1 & 3b \\ 1 & 4a \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \(\begin{vmatrix} 3b & b \\ 4a & a \end{vmatrix} = 3b \cdot a - b \cdot 4a = 3ab - 4ab = -ab\)

2. \(\begin{vmatrix} 1 & b \\ 1 & a \end{vmatrix} = 1 \cdot a - b \cdot 1 = a - b\)

3. \(\begin{vmatrix} 1 & 3b \\ 1 & 4a \end{vmatrix} = 1 \cdot 4a - 3b \cdot 1 = 4a - 3b\)

Substituting these back into the determinant:

\[ D = 1 \cdot (-ab) - 2a \cdot (a - b) + a \cdot (4a - 3b) \]

### Step 3: Simplify the determinant
Now we simplify:

\[ D = -ab - 2a(a - b) + a(4a - 3b) \]
\[ = -ab - 2a^2 + 2ab + 4a^2 - 3ab \]
\[ = (-ab + 2ab - 3ab) + (-2a^2 + 4a^2) \]
\[ = -2ab + 2a^2 \]

Setting the determinant equal to zero for concurrency:

\[ -2ab + 2a^2 = 0 \]

### Step 4: Factor the equation
Factoring out \(2a\):

\[ 2a(a - b) = 0 \]

### Step 5: Solve for \(a\) and \(b\)
This gives us two cases:

1. \(2a = 0 \Rightarrow a = 0\) (which is not valid since the lines must be distinct)
2. \(a - b = 0 \Rightarrow a = b\)

### Conclusion
Since \(a = b\), the point \((a, b)\) lies on the line \(y = x\), which is a straight line.