The distance between pair of parallel lines `9x^(2)-24xy+16y^(2)-12x+16y-12=0` is

Options

A. 5
B. 8
C. `8/5`
D. `5/8`

Answer

To find the distance between the pair of parallel lines given by the equation \(9x^2 - 24xy + 16y^2 - 12x + 16y - 12 = 0\), we can follow these steps:

### Step 1: Identify coefficients
The general form of a conic section is given by \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). From the given equation, we can identify:
- \(A = 9\)
- \(B = -24\)
- \(C = 16\)
- \(D = -12\)
- \(E = 16\)
- \(F = -12\)

### Step 2: Calculate \(g\), \(h\), and \(k\)
We need to calculate the values of \(g\), \(h\), and \(k\) for the formula to find the distance between the parallel lines:
- \(g = \frac{B}{2} = \frac{-24}{2} = -12\)
- \(h = \frac{D}{2} = \frac{-12}{2} = -6\)
- \(k = \frac{E}{2} = \frac{16}{2} = 8\)

### Step 3: Calculate \(g^2 - ac\)
Next, we need to calculate \(g^2 - ac\):
- \(g^2 = (-12)^2 = 144\)
- \(ac = A \cdot C = 9 \cdot 16 = 144\)
- Therefore, \(g^2 - ac = 144 - 144 = 0\)

### Step 4: Calculate the distance
The distance \(d\) between the parallel lines is given by the formula:
\[ d = \frac{2\sqrt{g^2 - ac}}{\sqrt{A} + \sqrt{C}} \]
Substituting the values we have:
\[ d = \frac{2\sqrt{0}}{\sqrt{9} + \sqrt{16}} = \frac{0}{3 + 4} = 0 \]

### Conclusion
The distance between the pair of parallel lines is \(0\).