Question 1165107
```python?code_reference&code_event_index=2
# Given information
h = 4  # x-coordinate of the vertex
k = 3  # y-coordinate of the vertex
latus_rectum_length = 4
condition = "4p > 0"

# The axis of symmetry is y = k (y = 3), so the parabola is horizontal,
# with the standard form: (y - k)^2 = 4p(x - h)

# The length of the latus rectum is |4p|.
# |4p| = 4.
# Since 4p > 0, we must have 4p = 4.
four_p = latus_rectum_length

# The equation is (y - k)^2 = 4p(x - h)
# Substitute the values
equation_lhs = f"(y - {k})^2"
equation_rhs = f"{four_p}(x - {h})"

print(f"Vertex (h, k): ({h}, {k})")
print(f"Axis of symmetry: y = {k}. (Horizontal parabola, form (y - k)^2 = 4p(x - h))")
print(f"Length of latus rectum |4p|: {latus_rectum_length}")
print(f"Condition: {condition}")
print(f"Since |4p| = {latus_rectum_length} and 4p > 0, we have 4p = {four_p}")
print(f"The equation of the parabola is: {equation_lhs} = {equation_rhs}")

```

```text?code_stdout&code_event_index=2
Vertex (h, k): (4, 3)
Axis of symmetry: y = 3. (Horizontal parabola, form (y - k)^2 = 4p(x - h))
Length of latus rectum |4p|: 4
Condition: 4p > 0
Since |4p| = 4 and 4p > 0, we have 4p = 4
The equation of the parabola is: (y - 3)^2 = 4(x - 4)

```

The equation of the parabola can be found using the given information:

1.  **Identify the Standard Form:**

      * The vertex is at $(h, k) = (4, 3)$.
      * The axis of symmetry is the horizontal line $y = 3$.
      * Since the axis of symmetry is horizontal ($y=k$), the parabola opens horizontally, and its standard form is:
        $$(y - k)^2 = 4p(x - h)$$

2.  **Substitute the Vertex Coordinates:**

      * Substituting $h=4$ and $k=3$ into the standard form gives:
        $$(y - 3)^2 = 4p(x - 4)$$

3.  **Determine the Value of $4p$ (Latus Rectum):**

      * The length of the latus rectum is $|4p|$.
      * We are given that the length of the latus rectum is $4$, so $|4p| = 4$.
      * This implies $4p = 4$ or $4p = -4$.
      * The condition $4p > 0$ tells us that we must choose the positive value:
        $$4p = 4$$

4.  **Final Equation:**

      * Substitute $4p = 4$ back into the equation from step 2:
        $$(y - 3)^2 = 4(x - 4)$$

The equation of the parabola is $\mathbf{(y - 3)^2 = 4(x - 4)}$.