To write the equation of the hyperbola given its center, vertex, and focus, we first identify some key information.
Key Information:
- Center: \((2, -5)\)
- Vertex: \((2, -2)\)
- Focus: \((2, -5 + 2\sqrt{3})\)
Step 1: Determine the Orientation Since the x-coordinates of the center, vertex, and focus are the same, the hyperbola opens vertically.
Step 2: Calculate \(a\) and \(c\)
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The distance from the center to the vertex (which is \(a\)) can be calculated: \[ a = |-5 - (-2)| = |-5 + 2| = 3 \]
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The distance from the center to the focus (which is \(c\)) is: \[ c = |-5 - (-5 + 2\sqrt{3})| = |2\sqrt{3}| \] Thus, \(c = 2\sqrt{3}\).
Step 3: Use the Relationship \(c^2 = a^2 + b^2\) Now we can find \(b\): \[ c^2 = a^2 + b^2 \] \[ (2\sqrt{3})^2 = 3^2 + b^2 \] \[ 12 = 9 + b^2 \] \[ b^2 = 12 - 9 = 3 \]
Step 4: Write the Equation The standard form of the equation for a vertically oriented hyperbola centered at \((h, k)\) is: \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \] Substituting \(h = 2\), \(k = -5\), \(a^2 = 9\), and \(b^2 = 3\): \[ \frac{(y + 5)^2}{9} - \frac{(x - 2)^2}{3} = 1 \]
Therefore, the correct equation of the hyperbola is: \[ \frac{(y + 5)^2}{9} - \frac{(x - 2)^2}{3} = 1 \]
Hence, the response is: Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 9 End Fraction minus Start Fraction left-parenthesis x minus 2 right-parenthesis squared over 3 End Fraction equals 1.