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Sagot :
Let's solve for the focus and directrix of the given parabola [tex]\( y = \frac{1}{2}(x+1)^2 + 4 \)[/tex].
First, we recognize that this equation is in the standard form of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola and [tex]\( a \)[/tex] is a constant.
For the given equation:
- [tex]\( a = \frac{1}{2} \)[/tex]
- [tex]\( h = -1 \)[/tex]
- [tex]\( k = 4 \)[/tex]
### 1. Finding the Focus:
The formula for the focus of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex] is [tex]\( \left( h, k + \frac{1}{4a} \right) \)[/tex].
Substituting the known values:
- [tex]\( h = -1 \)[/tex]
- [tex]\( k = 4 \)[/tex]
- [tex]\( a = \frac{1}{2} \)[/tex]
Compute [tex]\( \frac{1}{4a} \)[/tex]:
[tex]\[ \frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{2}} = \frac{1}{2} \][/tex]
Now, determine the coordinates of the focus:
[tex]\[ \text{Focus} = \left( h, k + \frac{1}{4a} \right) = \left( -1, 4 + \frac{1}{2} \right) = \left( -1, 4.5 \right) \][/tex]
### 2. Finding the Directrix:
The formula for the directrix of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex] is [tex]\( y = k - \frac{1}{4a} \)[/tex].
Substituting the known values:
- [tex]\( k = 4 \)[/tex]
- [tex]\( a = \frac{1}{2} \)[/tex]
Compute [tex]\( \frac{1}{4a} \)[/tex]:
[tex]\[ \frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{2}} = \frac{1}{2} \][/tex]
Now, determine the equation of the directrix:
[tex]\[ \text{Directrix} = y = k - \frac{1}{4a} = 4 - \frac{1}{2} = 3.5 \][/tex]
### Conclusion:
Given our focus and directrix calculations:
- Focus: [tex]\( \left( -1, 4.5 \right) \)[/tex]
- Directrix: [tex]\( y = 3.5 \)[/tex]
Let's compare these results with the provided options. The numerical values might seem off due to squaring and taking square roots out of context, leading to approximate values:
- Option A: Focus: [tex]\( \left( -1, 3^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 4^{1/2} \)[/tex]
- Option B: Focus: [tex]\( \left( -1, 4^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 3^{1/2} \)[/tex]
- Option C: Focus: [tex]\( \left( 1, 4^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 3^{1/2} \)[/tex]
- Option D: Focus: [tex]\( \left(1, 3^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 4^{1/2} \)[/tex]
The calculations previously confirmed:
- No options exactly match the correct numerical values for focus and directrix derived from the given equation.
Therefore, based on our analysis, none of the provided options (A, B, C, D) correctly match the true values of our calculations.
First, we recognize that this equation is in the standard form of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola and [tex]\( a \)[/tex] is a constant.
For the given equation:
- [tex]\( a = \frac{1}{2} \)[/tex]
- [tex]\( h = -1 \)[/tex]
- [tex]\( k = 4 \)[/tex]
### 1. Finding the Focus:
The formula for the focus of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex] is [tex]\( \left( h, k + \frac{1}{4a} \right) \)[/tex].
Substituting the known values:
- [tex]\( h = -1 \)[/tex]
- [tex]\( k = 4 \)[/tex]
- [tex]\( a = \frac{1}{2} \)[/tex]
Compute [tex]\( \frac{1}{4a} \)[/tex]:
[tex]\[ \frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{2}} = \frac{1}{2} \][/tex]
Now, determine the coordinates of the focus:
[tex]\[ \text{Focus} = \left( h, k + \frac{1}{4a} \right) = \left( -1, 4 + \frac{1}{2} \right) = \left( -1, 4.5 \right) \][/tex]
### 2. Finding the Directrix:
The formula for the directrix of a parabola [tex]\( y = a(x - h)^2 + k \)[/tex] is [tex]\( y = k - \frac{1}{4a} \)[/tex].
Substituting the known values:
- [tex]\( k = 4 \)[/tex]
- [tex]\( a = \frac{1}{2} \)[/tex]
Compute [tex]\( \frac{1}{4a} \)[/tex]:
[tex]\[ \frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{2}} = \frac{1}{2} \][/tex]
Now, determine the equation of the directrix:
[tex]\[ \text{Directrix} = y = k - \frac{1}{4a} = 4 - \frac{1}{2} = 3.5 \][/tex]
### Conclusion:
Given our focus and directrix calculations:
- Focus: [tex]\( \left( -1, 4.5 \right) \)[/tex]
- Directrix: [tex]\( y = 3.5 \)[/tex]
Let's compare these results with the provided options. The numerical values might seem off due to squaring and taking square roots out of context, leading to approximate values:
- Option A: Focus: [tex]\( \left( -1, 3^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 4^{1/2} \)[/tex]
- Option B: Focus: [tex]\( \left( -1, 4^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 3^{1/2} \)[/tex]
- Option C: Focus: [tex]\( \left( 1, 4^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 3^{1/2} \)[/tex]
- Option D: Focus: [tex]\( \left(1, 3^{1/2} \right) \)[/tex]; Directrix: [tex]\( y = 4^{1/2} \)[/tex]
The calculations previously confirmed:
- No options exactly match the correct numerical values for focus and directrix derived from the given equation.
Therefore, based on our analysis, none of the provided options (A, B, C, D) correctly match the true values of our calculations.
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