The lines `x+ay+a^(3)=0, x+by+b^(3)=0` and `x+cy+c^(3)=0` where a,b,c are all distinct are concurrent.

Options

A. for all values of a,b,c
B. if `a+b+c=0`
C. if `a^(3)+b^(3)+c^(3)-3abc=0`
D. for no values of a,b,c

Answer

To determine whether the lines \(x + ay + a^3 = 0\), \(x + by + b^3 = 0\), and \(x + cy + c^3 = 0\) are concurrent, we need to check if the determinant of the coefficients of these lines is equal to zero.

### Step 1: Write the equations in standard form
The given equations can be expressed in the form:
1. \(1x + ay + a^3 = 0\)
2. \(1x + by + b^3 = 0\)
3. \(1x + cy + c^3 = 0\)

### Step 2: Set up the determinant
The determinant of the coefficients can be set up as follows:

\[ \begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{vmatrix} \]

### Step 3: Calculate the determinant
To calculate the determinant, we can use the property of determinants that allows us to perform row operations. We will subtract the first row from the second and third rows:

\[ \begin{vmatrix} 1 & a & a^3 \\ 0 & b-a & b^3 - a^3 \\ 0 & c-a & c^3 - a^3 \end{vmatrix} \]

### Step 4: Simplify the determinant
Now we can simplify the determinant further. The determinant can be expanded as follows:

\[ = 1 \cdot \begin{vmatrix} b-a & b^3 - a^3 \\ c-a & c^3 - a^3 \end{vmatrix} \]

### Step 5: Factor the differences of cubes
Using the identity \(x^3 - y^3 = (x-y)(x^2 + xy + y^2)\), we can express \(b^3 - a^3\) and \(c^3 - a^3\):

\[ b^3 - a^3 = (b-a)(b^2 + ab + a^2) \]
\[ c^3 - a^3 = (c-a)(c^2 + ac + a^2) \]

Now our determinant becomes:

\[ = (b-a)(c-a) \cdot \begin{vmatrix} 1 & b^2 + ab + a^2 \\ 1 & c^2 + ac + a^2 \end{vmatrix} \]

### Step 6: Calculate the final determinant
Now we can calculate the 2x2 determinant:

\[ = (b-a)(c-a) \cdot \left[(b^2 + ab + a^2) - (c^2 + ac + a^2)\right] \]

This simplifies to:

\[ = (b-a)(c-a)(b^2 + ab + a^2 - c^2 - ac - a^2) \]

### Step 7: Set the determinant to zero
For the lines to be concurrent, this determinant must equal zero:

\[ (b-a)(c-a)(b^2 + ab - c^2 - ac) = 0 \]

### Step 8: Analyze the conditions
Since \(a\), \(b\), and \(c\) are distinct, \(b-a\) and \(c-a\) cannot be zero. Therefore, we must have:

\[ b^2 + ab - c^2 - ac = 0 \]

### Conclusion
From the above steps, we conclude that the lines are concurrent if:

\[ a + b + c = 0 \]