To solve this problem, let's break down the information provided and analyze each part of the question.

Part 1: Finding the Equation of the Parabola

1. Focus and Directrix: The focus is at $(-5, 5)$, and the directrix is the line $y = 3$.

2. Direction and Vertex:

   • A parabola with a vertical axis of symmetry opens upward or downward.
   • Since the focus is above the directrix, the parabola opens upward.

3. Vertex Calculation:

   • The vertex is located midway between the focus and the directrix.
   • Midpoint of the focus $(-5, 5)$ and the directrix $y = 3$: $\text{Vertex} = \left(-5, \frac{5 + 3}{2}\right) = \left(-5, 4\right)$

4. Distance from Vertex to Focus (p):

   • The distance $p$ is the vertical distance from the vertex to the focus. $p = 5 - 4 = 1$

5. Standard Form for Vertical Parabolas:

   • The equation for a vertical parabola with vertex $(h, k)$ is $(x - h)^2 = 4p(y - k)$.
   • Here, $(h, k) = (-5, 4)$ and $p = 1$.
   • Substituting these values gives: $(x + 5)^2 = 4 \cdot 1 \cdot (y - 4)$
   • Simplified: $(x + 5)^2 = 4(y - 4)$
   • This is the equation of the parabola.

Part 2: Finding the Points that Define the Latus Rectum

1. Latus Rectum Definition:

   • The latus rectum is a line segment perpendicular to the axis of symmetry and passes through the focus. Its length is $4p$.
   • Here, $p = 1$, so the length of the latus rectum is $4 \times 1 = 4$.

2. Points on the Latus Rectum:

   • Since the latus rectum is perpendicular to the axis of symmetry (the vertical line through the focus), it will be horizontal.
   • The focus is at $(-5, 5)$, so moving 2 units left and right from the focus gives the endpoints of the latus rectum.
   • These points are: $(-5 - 2, 5) = (-7, 5) \quad \text{and} \quad (-5 + 2, 5) = (-3, 5)$

Summary of Answers

• Equation of the Parabola: $(x + 5)^2 = 4(y - 4)$
• Points that Define the Latus Rectum: $(-7, 5)$ and $(-3, 5)$

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Related Questions

1. What is the general formula for a parabola given a focus and directrix?
2. How does the orientation of the parabola change if the focus is below the directrix?
3. How can we verify the vertex position using the midpoint formula?
4. What other conic sections use a focus and directrix in their definitions?
5. How does the value of $p$ affect the width of a parabola?

Tip

When working with parabolas, remember that the distance $p$ from the vertex to the focus also determines the length of the latus rectum as $4p$.