To determine the focus and directrix of the given parabola [tex]\( (x + 5)^2 = 16(y - 3) \)[/tex], we follow a systematic approach. The equation represents a vertical parabola (one that opens up or down).
Given the equation of the parabola:
[tex]\[ (x + 5)^2 = 16(y - 3) \][/tex]
Step 1: Identify the standard form of the parabola
The general form for a vertical parabola is:
[tex]\[ (x - h)^2 = 4p(y - k) \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\( p \)[/tex] is the distance from the vertex to the focus (and also from the vertex to the directrix).
Step 2: Rewrite the given equation to match the standard form
Our given equation:
[tex]\[ (x + 5)^2 = 16(y - 3) \][/tex]
Can be rewritten as:
[tex]\[ (x - (-5))^2 = 16(y - 3) \][/tex]
From this, we can identify:
[tex]\[ h = -5 \][/tex]
[tex]\[ k = 3 \][/tex]
[tex]\[ 4p = 16 \][/tex]
Step 3: Solve for [tex]\( p \)[/tex]
To find [tex]\( p \)[/tex]:
[tex]\[ 4p = 16 \][/tex]
[tex]\[ p = \frac{16}{4} \][/tex]
[tex]\[ p = 4 \][/tex]
Step 4: Determine the focus
Since the parabola opens upwards (because the coefficient of [tex]\( y \)[/tex] is positive):
- The focus is at a distance [tex]\( p \)[/tex] units above the vertex.
Given the vertex [tex]\((h, k) = (-5, 3)\)[/tex] and [tex]\( p = 4 \)[/tex]:
- The focus [tex]\((h, k + p)\)[/tex] is:
[tex]\[ \text{Focus} = (-5, 3 + 4) \][/tex]
[tex]\[ \text{Focus} = (-5, 7) \][/tex]
Step 5: Determine the directrix
The directrix is a horizontal line [tex]\( p \)[/tex] units below the vertex (for an upward opening parabola).
Given the vertex [tex]\((h, k) = (-5, 3)\)[/tex] and [tex]\( p = 4 \)[/tex]:
- The directrix [tex]\( y = k - p \)[/tex] is:
[tex]\[ \text{Directrix} = y = 3 - 4 \][/tex]
[tex]\[ \text{Directrix} = y = -1 \][/tex]
Summary:
- The focus of the given parabola is at [tex]\((-5, 7)\)[/tex].
- The directrix of the given parabola is [tex]\( y = -1 \)[/tex].
So, the answers are:
- Focus: [tex]\((-5, 7)\)[/tex]
- Directrix: [tex]\( y = -1 \)[/tex]
