The vertex and focus of parabola `(y-2)^(2) = -4(x-4)` are

Options

A. a) (4, 2), (3, 2)
B. b) (4. 2), (5. 2)
C. c) (4, 2), (4, 1)
D. d) (3. 2). (3. -2)

Answer

To find the vertex and focus of the parabola given by the equation \((y - 2)^2 = -4(x - 4)\), we can follow these steps:

### Step 1: Identify the standard form of the parabola
The given equation can be compared to the standard form of a parabola that opens to the left, which is:
\[ (y - k)^2 = -4a(x - h) \]
where \((h, k)\) is the vertex and \(a\) is the distance from the vertex to the focus.

### Step 2: Rewrite the equation
The equation we have is:
\[ (y - 2)^2 = -4(x - 4) \]
From this, we can identify:
- \(k = 2\)
- \(h = 4\)
- \(4a = 4\) (implying \(a = 1\))

### Step 3: Determine the vertex
The vertex \((h, k)\) can be directly found from the values we identified:
\[ \text{Vertex} = (h, k) = (4, 2) \]

### Step 4: Determine the focus
For a parabola that opens to the left, the focus is located at:
\[ (h - a, k) \]
Substituting the values we have:
\[ \text{Focus} = (4 - 1, 2) = (3, 2) \]

### Final Answer
Thus, the vertex of the parabola is \((4, 2)\) and the focus is \((3, 2)\).