The equation of the parabola with its vertex at (1, 1) and focus (3, 1) is

Options

A. `(x-1)^(2)=8(y-1)`
B. `(y-1)^(2)=8(x-3)`
C. `(y-1)^(2)=8(x-1)`
D. `(x-3)^(2)=8(y-1)`

Answer

To find the equation of the parabola with its vertex at (1, 1) and focus at (3, 1), we can follow these steps:

### Step 1: Identify the Vertex and Focus
- The vertex \( V \) of the parabola is given as \( (1, 1) \).
- The focus \( F \) of the parabola is given as \( (3, 1) \).

### Step 2: Determine the Orientation of the Parabola
- Since the y-coordinates of the vertex and focus are the same, the parabola opens horizontally (either to the right or left).
- Here, since the focus is to the right of the vertex, the parabola opens to the right.

### Step 3: Use the Standard Form of the Parabola
- The standard form of a horizontally opening parabola is given by:
\[ (y - k)^2 = 4p(x - h) \]
where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus.

### Step 4: Calculate the Value of \( p \)
- The distance \( p \) is the horizontal distance from the vertex to the focus.
- Since the vertex is at \( (1, 1) \) and the focus is at \( (3, 1) \):
\[ p = 3 - 1 = 2 \]

### Step 5: Substitute Values into the Standard Form
- Here, \( h = 1 \), \( k = 1 \), and \( p = 2 \).
- Substitute these values into the standard form:
\[ (y - 1)^2 = 4 \cdot 2 \cdot (x - 1) \]
Simplifying this gives:
\[ (y - 1)^2 = 8(x - 1) \]

### Step 6: Final Equation of the Parabola
- The equation of the parabola is:
\[ (y - 1)^2 = 8(x - 1) \]

### Summary
The equation of the parabola with its vertex at (1, 1) and focus at (3, 1) is:
\[ (y - 1)^2 = 8(x - 1) \]

---