The equation of the parabola whose vertex is at(2, -1) and focus at(2, -3), is

Options

A. `x^(2)+4x-8y-12=0`
B. `x^(2)-4x+8y+12=0`
C. `x^(2)+8y=12`
D. `x^(2)-4x+12=0`

Answer

To find the equation of the parabola with a vertex at (2, -1) and a focus at (2, -3), we can follow these steps:

### Step 1: Identify the vertex and focus
The vertex of the parabola is given as \( V(2, -1) \) and the focus is at \( F(2, -3) \).

### Step 2: Determine the orientation of the parabola
Since the x-coordinates of the vertex and focus are the same (both are 2), the parabola opens vertically. The focus is below the vertex, indicating that the parabola opens downwards.

### Step 3: Calculate the distance \( a \)
The distance \( a \) is the distance from the vertex to the focus. We can calculate it as follows:
\[ a = |y_{vertex} - y_{focus}| = |-1 - (-3)| = |-1 + 3| = |2| = 2 \]
Since the parabola opens downwards, we take \( a = -2 \).

### Step 4: Write the standard form of the parabola
The standard form of a vertically oriented parabola is given by:
\[ (y - k) = \frac{1}{4a}(x - h)^2 \]
where \( (h, k) \) is the vertex. Plugging in our values:
- \( h = 2 \)
- \( k = -1 \)
- \( a = -2 \)

The equation becomes:
\[ y + 1 = \frac{1}{4(-2)}(x - 2)^2 \]
This simplifies to:
\[ y + 1 = -\frac{1}{8}(x - 2)^2 \]

### Step 5: Rearranging the equation
To express this in a standard form, we can rearrange it:
\[ 8(y + 1) = -(x - 2)^2 \]
or
\[ -(x - 2)^2 + 8(y + 1) = 0 \]
This can be rewritten as:
\[ (x - 2)^2 + 8(y + 1) = 0 \]

### Step 6: Expand and simplify
Expanding the equation:
\[ (x - 2)^2 + 8y + 8 = 0 \]
\[ x^2 - 4x + 4 + 8y + 8 = 0 \]
Combining like terms gives:
\[ x^2 - 4x + 8y + 12 = 0 \]

### Final Answer
The equation of the parabola is:
\[ x^2 - 4x + 8y + 12 = 0 \]

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