Refer to the equation below. ( y + 5)² = - 1/2 ( x + 1 ) The length of latus rectum.
Group of answer choices

3/2 units

1/2 units

2/3 units

1/4 units

The given equation is:

\[
(y + 5)^2 = -\frac{1}{2}(x + 1)
\]

This equation resembles the standard form of the equation of a **parabola** that opens horizontally, which can be written as:

\[
(y - k)^2 = 4p(x - h)
\]

Here, \((h, k)\) is the vertex of the parabola, and \(p\) is the distance from the vertex to the focus (or the focal length).

Now let's rewrite the given equation to match the standard form:

\[
(y + 5)^2 = -\frac{1}{2}(x + 1)
\]

To match it to the standard form, we can rewrite it as:

\[
(y + 5)^2 = 2 \left(\frac{-1}{4} (x + 1)\right)
\]

So, comparing this with the standard form \((y - k)^2 = 4p(x - h)\), we have:

- \(h = -1\)
- \(k = -5\)
- \(4p = -\frac{1}{2}\)

Now, solving for \(p\):

\[
4p = -\frac{1}{2} \implies p = -\frac{1}{8}
\]

The length of the **latus rectum** of a parabola is given by \( |4p| \). So,

\[
|4p| = \left| -\frac{1}{2} \right| = \frac{1}{2}
\]

Thus, the length of the latus rectum is:

\[
\frac{1}{2} \, \text{units}
\]

The correct answer is **1/2 units**.

Would you like further clarification on any of the steps? Here are 5 related questions you might explore:

1. What is the geometric significance of the latus rectum in a parabola?
2. How can we identify whether a parabola opens horizontally or vertically from its equation?
3. What is the distance between the vertex and the focus in this parabola?
4. How does the value of \(p\) affect the shape of a parabola?
5. Can we determine the equation of the directrix from this parabola?

**Tip:** The latus rectum of a parabola passes through its focus and is always perpendicular to its axis of symmetry.

Refer to the equation below. ( y + 5)² = - 1/2 ( x + 1 ) The length of latus rectum. Group of answer choices: 3/2 units, 1/2 units, 2/3 units, 1/4 units.

Solve the problem of finding the length of the latus rectum for the parabola given by the equation (y + 5)² = -1/2 (x + 1). This involves using the standard form of a parabola and understanding the relationship between the vertex, focus, and the length of the latus rectum. Suitable for students in Grades 10-12.

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