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Conic Sections: Mastery Test

Use the drawing tools to form the correct answers on the graph. Plot the vertex, focus, and directrix for the parabola represented by this equation:

[tex]\[ x = -\frac{1}{12}(y-2)^2 + 1 \][/tex]

\begin{tabular}{|c|c|}
\hline Drawing Tools & Symbol \\
\hline Select & Q \\
\hline Point & [tex]$\bullet \quad 1$[/tex] \\
\hline
\end{tabular}

Click on a tool to begin drawing.


Sagot :

To plot the vertex, focus, and directrix for the parabola represented by the equation
[tex]\[ x = -\frac{1}{12}(y-2)^2 + 1 \][/tex]

we need to follow these steps:

### Step-by-Step Solution

1. Identify the vertex:
The equation of the parabola can be written in the form:
[tex]\[ x = a(y-k)^2 + h \][/tex]
Here, we have [tex]\( a = -\frac{1}{12} \)[/tex], [tex]\( k = 2 \)[/tex], and [tex]\( h = 1 \)[/tex]. The vertex [tex]\((h, k)\)[/tex] of the parabola is thus:
[tex]\[ \text{Vertex} = (1, 2) \][/tex]

2. Determine the value of [tex]\(p\)[/tex]:
The value of [tex]\(p\)[/tex] for a parabola of the form [tex]\( x = a(y-k)^2 + h \)[/tex] is given by:
[tex]\[ a = \frac{1}{4p} \][/tex]
Here, [tex]\( a = -\frac{1}{12} \)[/tex]. We can solve for [tex]\( p \)[/tex] by setting up the equation:
[tex]\[ -\frac{1}{12} = \frac{1}{4p} \][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[ -12 = 4p \quad \Rightarrow \quad p = -3 \][/tex]

3. Plot the focus:
The focus of the parabola is located at a distance of [tex]\( p \)[/tex] from the vertex along the axis of symmetry. Since [tex]\( p \)[/tex] is negative, the focus is to the left of the vertex. The coordinates of the focus are:
[tex]\[ \text{Focus} = \left( h + p, k \right) = (1 - 3, 2) = (-2, 2) \][/tex]

4. Determine the directrix:
The directrix is a vertical line perpendicular to the axis of symmetry of the parabola, and it is located at a distance of [tex]\( |p| \)[/tex] from the vertex on the side opposite the focus. The equation of the directrix is:
[tex]\[ x = h - p = 1 - (-3) = 1 + 3 = 4 \][/tex]

### Summary of Key Points to Plot:
- Vertex: [tex]\((1, 2)\)[/tex]
- Focus: [tex]\((-2, 2)\)[/tex]
- Directrix: [tex]\( x = 4 \)[/tex]

Using the drawing tools, you can plot these points and the directrix on the graph. The vertex is the point [tex]\((1, 2)\)[/tex], the focus is the point [tex]\((-2, 2)\)[/tex], and the directrix is the vertical line [tex]\( x = 4 \)[/tex].