Question
Rewrite the equation y squared minus 2 x plus 2 y minus 5 equals 0 in standard form. Determine the focus and directrix.
(1 point)
Responses
left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
Image with alt text: left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
left parenthesis y plus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
Image with alt text: left parenthesis y plus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
Image with alt text: left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
Image with alt text: left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
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(1 point)
Responses
left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
Image with alt text: left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
left parenthesis y plus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
Image with alt text: left parenthesis y plus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 7 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 5 over 2 end fraction
left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
Image with alt text: left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
Image with alt text: left parenthesis y minus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma negative 1 right parenthesis comma x equals negative start fraction 7 over 2 end fraction
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Answers
GPT-4o mini
To rewrite the equation \( y^2 - 2x + 2y - 5 = 0 \) in standard form, we can follow these steps:
1. **Isolate the \(y\) terms:**
\[
y^2 + 2y = 2x + 5
\]
2. **Complete the square for the \(y\) terms:**
The expression on the left can be completed as follows:
\[
y^2 + 2y + 1 = 2x + 5 + 1
\]
which gives us:
\[
(y + 1)^2 = 2x + 6
\]
Simplifying this, we get:
\[
(y + 1)^2 = 2(x + 3)
\]
Now, this is the standard form of a parabola:
\[
(y - k)^2 = 4p(x - h)
\]
where \((h, k)\) is the vertex. From the equation \((y + 1)^2 = 2(x + 3)\), we can identify:
- \(h = -3\)
- \(k = -1\)
- \(4p = 2\) or \(p = \frac{1}{2}\)
3. **Vertex**:
The vertex of the parabola is at \((-3, -1)\).
4. **Focus and Directrix**:
For a parabola that opens to the right:
- The focus is located at \((h + p, k)\):
\[
F = \left(-3 + \frac{1}{2}, -1 \right) = \left(-\frac{5}{2}, -1\right)
\]
- The directrix is a vertical line given by \(x = h - p\):
\[
x = -3 - \frac{1}{2} = -\frac{7}{2}
\]
So the final answers are:
\[
\text{Standard form: } (y + 1)^2 = 2(x + 3)
\]
\[
\text{Focus: } F = \left(-\frac{5}{2}, -1\right), \quad \text{Directrix: } x = -\frac{7}{2}
\]
Among the provided options, the correct answer is:
**left parenthesis y plus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma negative 1 right parenthesis , x equals negative start fraction 7 over 2 end fraction**.
1. **Isolate the \(y\) terms:**
\[
y^2 + 2y = 2x + 5
\]
2. **Complete the square for the \(y\) terms:**
The expression on the left can be completed as follows:
\[
y^2 + 2y + 1 = 2x + 5 + 1
\]
which gives us:
\[
(y + 1)^2 = 2x + 6
\]
Simplifying this, we get:
\[
(y + 1)^2 = 2(x + 3)
\]
Now, this is the standard form of a parabola:
\[
(y - k)^2 = 4p(x - h)
\]
where \((h, k)\) is the vertex. From the equation \((y + 1)^2 = 2(x + 3)\), we can identify:
- \(h = -3\)
- \(k = -1\)
- \(4p = 2\) or \(p = \frac{1}{2}\)
3. **Vertex**:
The vertex of the parabola is at \((-3, -1)\).
4. **Focus and Directrix**:
For a parabola that opens to the right:
- The focus is located at \((h + p, k)\):
\[
F = \left(-3 + \frac{1}{2}, -1 \right) = \left(-\frac{5}{2}, -1\right)
\]
- The directrix is a vertical line given by \(x = h - p\):
\[
x = -3 - \frac{1}{2} = -\frac{7}{2}
\]
So the final answers are:
\[
\text{Standard form: } (y + 1)^2 = 2(x + 3)
\]
\[
\text{Focus: } F = \left(-\frac{5}{2}, -1\right), \quad \text{Directrix: } x = -\frac{7}{2}
\]
Among the provided options, the correct answer is:
**left parenthesis y plus 1 right parenthesis squared equals 2 left parenthesis x plus 3 right parenthesis line break upper F equals left parenthesis negative start fraction 5 over 2 end fraction comma negative 1 right parenthesis , x equals negative start fraction 7 over 2 end fraction**.