Answer :
To find the equation of the parabola with the given focus and directrix, we first recognize that this is a case of a horizontal parabola.
### Given:
- Focus: [tex]\((0, 3)\)[/tex]
- Directrix: [tex]\(x = -2\)[/tex]
### Step-by-Step Solution:
1. Identify the Coordinates [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
The vertex form of a horizontal parabola is [tex]\((y - k)^2 = 4a(x - h)\)[/tex].
- Here, the focus [tex]\( (h + a, k) \)[/tex] is [tex]\((0 + a, 3)\)[/tex] which simplifies to [tex]\((a, 3)\)[/tex].
- The vertex [tex]\((h, k)\)[/tex] is [tex]\((0, 3)\)[/tex].
2. Determine [tex]\( a \)[/tex] Using Directrix:
The directrix equation for a horizontal parabola is [tex]\(x = h - a\)[/tex].
- Given directrix is [tex]\(x = -2\)[/tex].
- Plugging in the vertex [tex]\(h = 0\)[/tex]: [tex]\(0 - a = -2\)[/tex] which simplifies to [tex]\(a = 2\)[/tex].
3. Verify:
Since the focus is [tex]\((0 + 2, 3)\)[/tex] and [tex]\( a = 2 \)[/tex], indeed, the directrix [tex]\(x = 0 - 2 = -2\)[/tex] is correct. However, since we chose a towards left direction from vertex, [tex]\( a \)[/tex] turns out to be [tex]\(-2\)[/tex].
4. Formulate the Standard Equation of the Parabola:
Using the standard equation [tex]\((y - k)^2 = 4a(x - h)\)[/tex]:
- Substituting [tex]\(k = 3\)[/tex], [tex]\(h = 0\)[/tex], and [tex]\(a = -2\)[/tex]:
[tex]\[ (y - 3)^2 = 4(-2)(x - 0) \][/tex]
Simplifying, we get:
[tex]\[ (y - 3)^2 = -8x \][/tex]
### Final Parabola Equation:
The equation of the parabola with focus [tex]\((0, 3)\)[/tex] and directrix [tex]\(x = -2\)[/tex] is:
[tex]\[ (y - 3)^2 = -8x \][/tex]
### Given:
- Focus: [tex]\((0, 3)\)[/tex]
- Directrix: [tex]\(x = -2\)[/tex]
### Step-by-Step Solution:
1. Identify the Coordinates [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
The vertex form of a horizontal parabola is [tex]\((y - k)^2 = 4a(x - h)\)[/tex].
- Here, the focus [tex]\( (h + a, k) \)[/tex] is [tex]\((0 + a, 3)\)[/tex] which simplifies to [tex]\((a, 3)\)[/tex].
- The vertex [tex]\((h, k)\)[/tex] is [tex]\((0, 3)\)[/tex].
2. Determine [tex]\( a \)[/tex] Using Directrix:
The directrix equation for a horizontal parabola is [tex]\(x = h - a\)[/tex].
- Given directrix is [tex]\(x = -2\)[/tex].
- Plugging in the vertex [tex]\(h = 0\)[/tex]: [tex]\(0 - a = -2\)[/tex] which simplifies to [tex]\(a = 2\)[/tex].
3. Verify:
Since the focus is [tex]\((0 + 2, 3)\)[/tex] and [tex]\( a = 2 \)[/tex], indeed, the directrix [tex]\(x = 0 - 2 = -2\)[/tex] is correct. However, since we chose a towards left direction from vertex, [tex]\( a \)[/tex] turns out to be [tex]\(-2\)[/tex].
4. Formulate the Standard Equation of the Parabola:
Using the standard equation [tex]\((y - k)^2 = 4a(x - h)\)[/tex]:
- Substituting [tex]\(k = 3\)[/tex], [tex]\(h = 0\)[/tex], and [tex]\(a = -2\)[/tex]:
[tex]\[ (y - 3)^2 = 4(-2)(x - 0) \][/tex]
Simplifying, we get:
[tex]\[ (y - 3)^2 = -8x \][/tex]
### Final Parabola Equation:
The equation of the parabola with focus [tex]\((0, 3)\)[/tex] and directrix [tex]\(x = -2\)[/tex] is:
[tex]\[ (y - 3)^2 = -8x \][/tex]
