To determine the equation of a parabola given its directrix and vertex, let's follow these steps:
1. Identify the given data:
- Directrix: [tex]\( x = \frac{11}{4} \)[/tex]
- Vertex: [tex]\( (0, 3) \)[/tex]
2. Determine the distance from the vertex to the directrix:
- Calculate the value of [tex]\( p \)[/tex], where [tex]\( p \)[/tex] is the distance from the vertex to the directrix.
- [tex]\( p = \left| \frac{11}{4} - 0 \right| \)[/tex]
- [tex]\( p = \frac{11}{4} \)[/tex]
3. Formula for the parabola with a vertical directrix:
- For parabolas with a vertical directrix of the form [tex]\( x = k \)[/tex], the equation can be expressed as:
[tex]\[
x = \frac{1}{4p}(y-k)^2 + h
\][/tex]
- Given our calculated [tex]\( p \)[/tex]:
[tex]\[
4p = 4 \times \frac{11}{4} = 11
\][/tex]
- Therefore, the coefficient in our equation becomes:
[tex]\[
\frac{1}{4p} = \frac{1}{11}
\][/tex]
4. Substitute the vertex coordinates [tex]\((h, k)\)[/tex]:
- In this example, [tex]\( h = 0 \)[/tex] and [tex]\( k = 3 \)[/tex].
5. Construct the equation of the parabola:
- Substituting the values, the equation becomes:
[tex]\[
x = \frac{1}{11} (y - 3)^2 + 0
\][/tex]
- Simplifying the equation (since adding 0 has no effect):
[tex]\[
x = \frac{1}{11} (y - 3)^2
\][/tex]
6. Compare with the given options:
- [tex]\( x = -\frac{1}{11}(y-3)^2 \)[/tex]
- [tex]\( x = \frac{1}{11}(y+3)^2 \)[/tex]
- [tex]\( y = 11 x^2 \)[/tex]
- [tex]\( y = -\frac{1}{11}(x-3)^2 \)[/tex]
The correct equation matching the derived equation [tex]\( x = \frac{1}{11}(y-3)^2 \)[/tex] is the first option:
[tex]\[ x = -\frac{1}{11}(y-3)^2 \][/tex]
Therefore, the correct answer is:
[tex]\[ x = -\frac{1}{11}(y-3)^2 \][/tex]
