To find the equations of the parabolas given the focus and directrix, we will follow a systematic approach for each part of the question.
### Part (i): Focus at (5, 0) and Directrix x = -5
1. **Identify the Focus and Directrix**:
- Focus (F) = (5, 0)
- Directrix (D) = x = -5
2. **Determine the Vertex**:
- The vertex (V) of the parabola lies halfway between the focus and the directrix.
- The x-coordinate of the vertex can be found as:
\[
V_x = \frac{5 + (-5)}{2} = \frac{0}{2} = 0
\]
- The y-coordinate of the vertex is the same as the focus since the directrix is vertical:
\[
V_y = 0
\]
- Therefore, the vertex is at (0, 0).
3. **Determine the Orientation**:
- Since the focus is to the right of the directrix, the parabola opens to the right.
4. **Use the Standard Form of the Parabola**:
- The standard form of a parabola that opens to the right is given by:
\[
(y - k)^2 = 4p(x - h)
\]
- Here, (h, k) is the vertex, and p is the distance from the vertex to the focus.
- The distance p can be calculated as:
\[
p = 5 - 0 = 5
\]
5. **Substituting Values**:
- Substituting h = 0, k = 0, and p = 5 into the standard form:
\[
(y - 0)^2 = 4 \times 5 (x - 0)
\]
- This simplifies to:
\[
y^2 = 20x
\]
### Equation for Part (i):
\[
y^2 = 20x
\]
---
### Part (ii): Vertex at (1, 2) and Directrix x + y + 1 = 0
1. **Identify the Vertex**:
- Vertex (V) = (1, 2)
2. **Determine the Directrix**:
- The equation of the directrix is given as x + y + 1 = 0.
- We can rewrite this in slope-intercept form:
\[
y = -x - 1
\]
- This line has a slope of -1.
3. **Find the Distance from the Vertex to the Directrix**:
- The distance from a point (x_0, y_0) to a line Ax + By + C = 0 is given by:
\[
\text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
- For the line x + y + 1 = 0, A = 1, B = 1, C = 1, and the vertex (1, 2):
\[
\text{Distance} = \frac{|1 \cdot 1 + 1 \cdot 2 + 1|}{\sqrt{1^2 + 1^2}} = \frac{|1 + 2 + 1|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}
\]
4. **Determine the Orientation**:
- Since the directrix is not vertical, we need to find the direction in which the parabola opens.
- The parabola opens away from the directrix.
5. **Use the Standard Form of the Parabola**:
- The general form for a parabola that opens in the direction of the vertex is:
\[
(x - h)^2 = 4p(y - k)
\]
- Here, (h, k) is the vertex, and p is the distance from the vertex to the directrix.
- Since the distance is 2√2, we have p = 2√2.
6. **Substituting Values**:
- Substituting h = 1, k = 2, and p = 2√2 into the standard form:
\[
(x - 1)^2 = 4 \times 2\sqrt{2}(y - 2)
\]
- This simplifies to:
\[
(x - 1)^2 = 8\sqrt{2}(y - 2)
\]
### Equation for Part (ii):
\[
(x - 1)^2 = 8\sqrt{2}(y - 2)
\]
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