If 3x+4y+k=0 represents the equation of tangent at the vertex of the parabola `16x^(2)-24xy+9y^2+14x+2y+7=0`, then the value of k is ________ .

Explanation

To solve the problem step by step, we need to find the value of \( k \) such that the line \( 3x + 4y + k = 0 \) is the tangent at the vertex of the given parabola \( 16x^2 - 24xy + 9y^2 + 14x + 2y + 7 = 0 \).

### Step 1: Identify the parabola's equation
The given equation of the parabola is:
\[ 16x^2 - 24xy + 9y^2 + 14x + 2y + 7 = 0 \]

### Step 2: Rewrite the parabola in standard form
To find the vertex, we can rewrite the equation in a more manageable form. We can group the terms:
\[ 16x^2 - 24xy + 9y^2 + 14x + 2y + 7 = 0 \]

### Step 3: Find the coefficients
From the equation, we can identify the coefficients:
- \( A = 16 \)
- \( B = -24 \)
- \( C = 9 \)
- \( D = 14 \)
- \( E = 2 \)
- \( F = 7 \)

### Step 4: Calculate the vertex of the parabola
The vertex \( (h, k) \) of the parabola can be found using the formula:
\[ h = \frac{BD - 2AE}{A^2 - AB + C^2} \]
\[ k = \frac{CD - 2AF}{A^2 - AB + C^2} \]

Calculating \( A^2 - AB + C^2 \):
\[ A^2 = 16^2 = 256, \quad AB = 16 \times (-24) = -384, \quad C^2 = 9^2 = 81 \]
\[ A^2 - AB + C^2 = 256 + 384 + 81 = 721 \]

Now calculating \( h \):
\[ h = \frac{(-24)(14) - 2(16)(2)}{721} = \frac{-336 - 64}{721} = \frac{-400}{721} \]

Now calculating \( k \):
\[ k = \frac{(9)(14) - 2(16)(7)}{721} = \frac{126 - 224}{721} = \frac{-98}{721} \]

### Step 5: Find the equation of the axis of the parabola
The axis of the parabola can be found using the formula:
\[ Ax + By + C = 0 \implies 4x - 3y + \lambda = 0 \]

### Step 6: Set up the tangent equation
The tangent line at the vertex can be expressed as:
\[ 3x + 4y + k = 0 \]

### Step 7: Compare coefficients
From the equations, we can compare coefficients:
1. \( 8\lambda - 3\mu = 14 \)
2. \( 6\lambda + 4\mu = 2 \)

### Step 8: Solve the system of equations
From the first equation:
\[ 8\lambda - 3\mu = 14 \quad \text{(1)} \]
From the second equation:
\[ 6\lambda + 4\mu = 2 \quad \text{(2)} \]

Multiply equation (1) by 2:
\[ 16\lambda - 6\mu = 28 \quad \text{(3)} \]

Multiply equation (2) by 3:
\[ 18\lambda + 12\mu = 6 \quad \text{(4)} \]

Now add equations (3) and (4):
\[ (16\lambda - 6\mu) + (18\lambda + 12\mu) = 28 + 6 \]
\[ 34\lambda + 6\mu = 34 \]
\[ \lambda = 1 \]

### Step 9: Substitute to find \( \mu \)
Substituting \( \lambda = 1 \) back into equation (1):
\[ 8(1) - 3\mu = 14 \implies 8 - 3\mu = 14 \implies -3\mu = 6 \implies \mu = -2 \]

### Step 10: Find \( k \)
Using the constant part:
\[ \lambda^2 - \mu k = 7 \]
Substituting \( \lambda = 1 \) and \( \mu = -2 \):
\[ 1^2 - (-2)k = 7 \implies 1 + 2k = 7 \implies 2k = 6 \implies k = 3 \]

### Final Answer
The value of \( k \) is \( \boxed{3} \).