The separate equations of the lines given by `(x-4)^(2)-16(y-5)^(2)=0` are

Options

A. `x+4y+16=0 and x-4y-24=0`
B. `x+4y-16=0 and x-4y-24=0`
C. `x+4y-16=0 and x-4y+24=0`
D. `x+4y+16=0 and x-4y+24=0`

Answer

To solve the problem of finding the separate equations of the lines given by the equation \((x-4)^{2} - 16(y-5)^{2} = 0\), we can follow these steps:

### Step 1: Rewrite the given equation
Start with the given equation:
\[ (x - 4)^{2} - 16(y - 5)^{2} = 0 \]

### Step 2: Recognize the difference of squares
This equation can be recognized as a difference of squares, which can be factored as:
\[ A^{2} - B^{2} = (A - B)(A + B) \]
where \(A = (x - 4)\) and \(B = 4(y - 5)\).

### Step 3: Factor the equation
Using the difference of squares, we can factor the equation:
\[ (x - 4) - 4(y - 5) = 0 \quad \text{and} \quad (x - 4) + 4(y - 5) = 0 \]

### Step 4: Simplify the factors
Now, simplify each factor:
1. For the first factor:
\[ x - 4 - 4y + 20 = 0 \implies x - 4y + 16 = 0 \]
This can be rewritten as:
\[ x - 4y + 16 = 0 \]

2. For the second factor:
\[ x - 4 + 4y - 20 = 0 \implies x + 4y - 24 = 0 \]
This can be rewritten as:
\[ x + 4y - 24 = 0 \]

### Step 5: Write the separate equations
The separate equations of the lines are:
1. \(x - 4y + 16 = 0\)
2. \(x + 4y - 24 = 0\)

### Final Result
Thus, the separate equations of the lines are:
\[ \begin{align*} 1. & \quad x - 4y + 16 = 0 \\ 2. & \quad x + 4y - 24 = 0 \end{align*} \]

---