The equation of the directrix of the parabola `25{(x-2)^(2)+(y+5)^(2)}=(3x+4y-1)^(2),` is

Options

A. 3x+4y=0
B. 3x+4y-1=0
C. 4x-3y=0
D. 3x+4y+1=0

Explanation

To find the equation of the directrix of the given parabola \( 25(x - 2)^2 + (y + 5)^2 = (3x + 4y - 1)^2 \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten by dividing both sides by 25:
\[ (x - 2)^2 + \frac{(y + 5)^2}{25} = \frac{(3x + 4y - 1)^2}{25} \]

### Step 2: Identify the standard form
The equation now resembles the standard form of a parabola where:
\[ SP = PM \]
Here, \( S \) is the focus and \( P \) is any point on the parabola, while \( M \) is the foot of the perpendicular from point \( P \) to the directrix.

### Step 3: Determine the focus
From the equation, we can identify the focus \( S \) of the parabola as:
\[ S = (2, -5) \]

### Step 4: Identify the directrix
The directrix is given by the line equation \( 3x + 4y - 1 = 0 \). This line can be rearranged to find the directrix in slope-intercept form if necessary, but it is sufficient to express it in its current form.

### Step 5: Conclusion
Thus, the equation of the directrix of the parabola is:
\[ 3x + 4y - 1 = 0 \]