Question

13 of 1613 of 16 Items











Question
Write the equation of the hyperbola with center left-parenthesis 2 comma negative 5 right-parenthesis, vertex left-parenthesis 2 comma negative 2 right-parenthesis, and focus left-parenthesis 2 comma negative 5 plus 2 Start Root 3 End Root right-parenthesis.
(1 point)
Responses

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
Image with alt text: 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

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 12 End Fraction equals 1
Image with alt text: 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 12 End Fraction equals 1

Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 3 End Fraction minus Start Fraction left-parenthesis x minus 2 right-parenthesis squared over 9 End Fraction equals 1
Image with alt text: Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 3 End Fraction minus Start Fraction left-parenthesis x minus 2 right-parenthesis squared over 9 End Fraction equals 1

Start Fraction left-parenthesis y plus 5 right-parenthesis squared over 12 End Fraction minus Start Fraction left-parenthesis x minus 2 right-parenthesis squared over 9 End Fraction equals 1

Answer 1

To write the equation of the hyperbola given its center, vertex, and focus, we first identify some key information.

**Key Information:**
1. **Center**: \((2, -5)\)
2. **Vertex**: \((2, -2)\)
3. **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 \(c\)**
- The distance from the center to the vertex (which is \(a\)) can be calculated:
\[
a = |-5 - (-2)| = |-5 + 2| = 3
\]

- 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**.