To rewrite the equation \( y^2 - 2x + 2y - 5 = 0 \) in standard form, we can follow these steps:
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Isolate the \(y\) terms: \[ y^2 + 2y = 2x + 5 \]
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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 \(p = \frac{1}{2}\)
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Vertex: The vertex of the parabola is at \((-3, -1)\).
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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.
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