If the vertex and focus of a parabola are `(3,3)` and `(-3,3)` respectively, then its equation is

Options

A. A. `x^2-6x+24y-63=0`
B. B. `x^2-6x+24y-81=0`
C. C. `y^2-6y+24x-63=0`
D. D.`y^2-6y-24x+81=0`

Answer

To find the equation of the parabola given its vertex and focus, we can follow these steps:

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

### Step 2: Determine the Orientation of the Parabola
Since the y-coordinates of the vertex and focus are the same (both are 3), the parabola opens horizontally. The general form for a horizontally opening parabola is:
\[ (y - k)^2 = 4p(x - h) \]
where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus.

### Step 3: Calculate the Distance \( p \)
The distance \( p \) can be calculated as the distance between the vertex and the focus. Since the vertex is at \( (3, 3) \) and the focus is at \( (-3, 3) \):
\[ p = x_{focus} - x_{vertex} = -3 - 3 = -6 \]
Thus, \( p = -6 \).

### Step 4: Substitute Values into the Parabola Equation
Now we can substitute \( h = 3 \), \( k = 3 \), and \( p = -6 \) into the equation:
\[ (y - 3)^2 = 4(-6)(x - 3) \]
This simplifies to:
\[ (y - 3)^2 = -24(x - 3) \]

### Step 5: Expand the Equation
Now we will expand the equation:
\[ (y - 3)^2 = -24x + 72 \]
Expanding the left side:
\[ y^2 - 6y + 9 = -24x + 72 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ y^2 - 6y + 24x + 9 - 72 = 0 \]
This simplifies to:
\[ y^2 - 6y + 24x - 63 = 0 \]

### Conclusion
The equation of the parabola is:
\[ y^2 - 6y + 24x - 63 = 0 \]