The lines `x=3,y=4and 4x-3y+a=0` are concurrent for a equal to

Options

A. 0
B. -1
C. 2
D. 3

Answer

To determine the value of \( a \) for which the lines \( x = 3 \), \( y = 4 \), and \( 4x - 3y + a = 0 \) are concurrent, we can follow these steps:

### Step 1: Write the equations in standard form
The equations can be rewritten in the standard form:
1. \( x - 3 = 0 \)
2. \( y - 4 = 0 \)
3. \( 4x - 3y + a = 0 \)

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For the first line \( x - 3 = 0 \):
- Coefficient of \( x \) (let's call it \( c_1 \)) = 1
- Coefficient of \( y \) (let's call it \( d_1 \)) = 0
- Constant term (let's call it \( e_1 \)) = -3

- For the second line \( y - 4 = 0 \):
- Coefficient of \( x \) (let's call it \( c_2 \)) = 0
- Coefficient of \( y \) (let's call it \( d_2 \)) = 1
- Constant term (let's call it \( e_2 \)) = -4

- For the third line \( 4x - 3y + a = 0 \):
- Coefficient of \( x \) (let's call it \( c_3 \)) = 4
- Coefficient of \( y \) (let's call it \( d_3 \)) = -3
- Constant term (let's call it \( e_3 \)) = \( a \)

### Step 3: Set up the determinant
For the three lines to be concurrent, the determinant of the coefficients must be zero:
\[ \begin{vmatrix} c_1 & d_1 & e_1 \\ c_2 & d_2 & e_2 \\ c_3 & d_3 & e_3 \end{vmatrix} = 0 \]
Substituting the values we found:
\[ \begin{vmatrix} 1 & 0 & -3 \\ 0 & 1 & -4 \\ 4 & -3 & a \end{vmatrix} = 0 \]

### Step 4: Calculate the determinant
Calculating the determinant:
\[ = 1 \cdot \begin{vmatrix} 1 & -4 \\ -3 & a \end{vmatrix} - 0 + (-3) \cdot \begin{vmatrix} 0 & 1 \\ 4 & -3 \end{vmatrix} \]
Calculating the 2x2 determinants:
1. \( \begin{vmatrix} 1 & -4 \\ -3 & a \end{vmatrix} = 1 \cdot a - (-4)(-3) = a - 12 \)

2. \( \begin{vmatrix} 0 & 1 \\ 4 & -3 \end{vmatrix} = 0 \cdot (-3) - 1 \cdot 4 = -4 \)

Putting it all together:
\[ = 1(a - 12) - 0 - 3(-4) = a - 12 + 12 = a \]

### Step 5: Set the determinant to zero
Setting the determinant equal to zero for concurrency:
\[ a = 0 \]

### Conclusion
Thus, the value of \( a \) for which the lines are concurrent is:
\[ \boxed{0} \]