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Question 1165107: Find the equation of the parabola given vertex at (4, 3), axis of symmetry 𝑦 = 3, length of the latus rectum is 4, and 4𝑝 > 0
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! ```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 $\mathbf{(y - 3)^2 = 4(x - 4)}$.
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