To determine the focus, directrix, focal diameter, vertex, and axis of symmetry for the given parabola [tex]\( 26.8 y = x^2 \)[/tex], let's break it down step-by-step.
### Step 1: Understand the given equation
The given equation is:
[tex]\[ 26.8 y = x^2 \][/tex]
To analyze it, let's rewrite it in a more standard form. Divide both sides by 26.8:
[tex]\[ y = \frac{x^2}{26.8} \][/tex]
This equation represents a parabola that opens upwards, and its standard form can be compared to:
[tex]\[ y = kx^2 \][/tex]
### Step 2: Rewrite in standard parabola form
Rewrite the equation as:
[tex]\[ x^2 = 26.8y \][/tex]
Compare this with the general form of a parabola [tex]\( (x - h)^2 = 4a(y - k) \)[/tex]:
[tex]\[ x^2 = 4a (y - 0) \][/tex]
From this comparison, it is clear that:
[tex]\[ 4a = 26.8 \][/tex]
[tex]\[ a = \frac{26.8}{4} \][/tex]
[tex]\[ a = 6.7 \][/tex]
### Step 3: Identify the components
#### Vertex
The vertex (h, k) can be directly read from the equation's standard form [tex]\( (x - h)^2 = 4a (y - k) \)[/tex]. For this parabola:
[tex]\[ h = 0 \][/tex]
[tex]\[ k = 0 \][/tex]
Thus, the vertex is:
[tex]\[ \text{Vertex} = (0, 0) \][/tex]
#### Axis of Symmetry
The axis of symmetry for a parabola [tex]\( (x - h)^2 = 4a (y - k) \)[/tex] is [tex]\( x = h \)[/tex]. In this case:
[tex]\[ \text{Axis of Symmetry} = x = 0 \][/tex]
#### Focus
The focus of the parabola [tex]\( (x - h)^2 = 4a (y - k) \)[/tex] is at the point [tex]\( (h, k + a) \)[/tex]. Here, we have:
[tex]\[ h = 0 \][/tex]
[tex]\[ k = 0 \][/tex]
[tex]\[ a = 6.7 \][/tex]
Thus, the focus is:
[tex]\[ \text{Focus} = (0, 6.7) \][/tex]
#### Directrix
The directrix of the parabola [tex]\( (x - h)^2 = 4a (y - k) \)[/tex] is the line [tex]\( y = k - a \)[/tex]. Here:
[tex]\[ k = 0 \][/tex]
[tex]\[ a = 6.7 \][/tex]
Thus, the directrix is:
[tex]\[ \text{Directrix} = y = -6.7 \][/tex]
#### Focal Diameter
The focal diameter of the parabola is [tex]\( |4a| \)[/tex]. Here:
[tex]\[ a = 6.7 \][/tex]
[tex]\[ \text{Focal Diameter} = |4a| = |4 \cdot 6.7| = 26.8 \][/tex]
### Summary
- The focus is [tex]\((0, 6.7)\)[/tex].
- The directrix is [tex]\( y = -6.7 \)[/tex].
- The focal diameter is [tex]\( 26.8 \)[/tex].
- The vertex is [tex]\((0, 0)\)[/tex].
- The axis of symmetry is [tex]\( x = 0 \)[/tex].
Thus, the final answers are:
- The focus is [tex]\( (0, 6.7) \)[/tex]
- The directrix is [tex]\( y = -6.7 \)[/tex]
- The focal diameter is [tex]\( 26.8 \)[/tex]
- The vertex is [tex]\( (0, 0) \)[/tex]
- The axis of symmetry is [tex]\( x = 0 \)[/tex]
