The parabola `(y+1)^(2)=a(x-2)` passes through point (1,-2). The equaiton of its directrix is

Options

A. a) 4x + 1 = 0
B. b) 4x -1 = 0
C. c) 4x + 9 = 0
D. d) 4x - 9 = 0

Answer

To find the equation of the directrix of the parabola given by \((y + 1)^2 = a(x - 2)\) that passes through the point (1, -2), we can follow these steps:

### Step 1: Substitute the point into the parabola equation
We know the parabola passes through the point (1, -2). We can substitute \(x = 1\) and \(y = -2\) into the equation:
\[ (-2 + 1)^2 = a(1 - 2) \]
This simplifies to:
\[ (-1)^2 = a(-1) \]
\[ 1 = -a \]

### Step 2: Solve for \(a\)
From the equation \(1 = -a\), we can solve for \(a\):
\[ a = -1 \]

### Step 3: Rewrite the parabola equation
Now that we have \(a\), we can rewrite the equation of the parabola:
\[ (y + 1)^2 = -1(x - 2) \]
This can be rearranged to:
\[ (y + 1)^2 = -x + 2 \]
or
\[ (y + 1)^2 = -1(x - 2) \]

### Step 4: Identify the standard form of the parabola
The standard form of a parabola that opens left or right is:
\[ (y - k)^2 = 4p(x - h) \]
where \((h, k)\) is the vertex of the parabola. From our equation, we can identify:
- \(h = 2\)
- \(k = -1\)
- \(4p = -1\)

### Step 5: Solve for \(p\)
From \(4p = -1\), we can find \(p\):
\[ p = -\frac{1}{4} \]

### Step 6: Find the directrix
The directrix of a parabola is given by the equation:
\[ x = h - p \]
Substituting the values we found:
\[ x = 2 - \left(-\frac{1}{4}\right) \]
This simplifies to:
\[ x = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]

### Step 7: Write the equation of the directrix
Thus, the equation of the directrix is:
\[ x = \frac{9}{4} \]
This can also be expressed in standard form:
\[ 4x - 9 = 0 \]

### Final Answer
The equation of the directrix is:
\[ 4x - 9 = 0 \]