To find the coordinates of the center, foci, endpoints of the latus rectum, and the lengths of the axes of the hyperbola given by the equation \(9x^2 - 16y^2 + 18x - 64y - 199 = 0\), we will follow these steps:
### Step 1: Rearranging the Equation
Start by rearranging the equation into a standard form.
Given:
\[
9x^2 - 16y^2 + 18x - 64y - 199 = 0
\]
Rearranging gives:
\[
9x^2 + 18x - 16y^2 - 64y = 199
\]
### Step 2: Completing the Square
Next, we will complete the square for the \(x\) and \(y\) terms.
For \(x\):
\[
9(x^2 + 2x) = 9((x + 1)^2 - 1) = 9(x + 1)^2 - 9
\]
For \(y\):
\[
-16(y^2 + 4y) = -16((y + 2)^2 - 4) = -16(y + 2)^2 + 64
\]
Substituting these back into the equation:
\[
9(x + 1)^2 - 9 - 16(y + 2)^2 + 64 = 199
\]
\[
9(x + 1)^2 - 16(y + 2)^2 + 55 = 199
\]
\[
9(x + 1)^2 - 16(y + 2)^2 = 144
\]
### Step 3: Dividing by 144
Now, divide the entire equation by 144 to get it into standard form:
\[
\frac{(x + 1)^2}{16} - \frac{(y + 2)^2}{9} = 1
\]
### Step 4: Identifying Parameters
From the standard form \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\), we can identify:
- Center \((h, k) = (-1, -2)\)
- \(a^2 = 16 \Rightarrow a = 4\)
- \(b^2 = 9 \Rightarrow b = 3\)
### Step 5: Finding the Foci
The foci of the hyperbola are given by:
\[
(h \pm ae, k)
\]
where \(e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4}\).
Calculating the foci:
\[
Foci = (-1 \pm 4 \cdot \frac{5}{4}, -2) = (-1 \pm 5, -2)
\]
Thus, the foci are:
\[
(-6, -2) \quad \text{and} \quad (4, -2)
\]
### Step 6: Finding the Endpoints of the Latus Rectum
The endpoints of the latus rectum are given by:
\[
(h \pm ae, k \pm b)
\]
Calculating the endpoints:
\[
Endpoints = (-1 \pm 5, -2 \pm 3)
\]
This gives:
1. \((-6, 1)\)
2. \((-6, -5)\)
3. \((4, 1)\)
4. \((4, -5)\)
### Step 7: Length of the Axes
The lengths of the axes are:
- Major axis length = \(2a = 8\)
- Minor axis length = \(2b = 6\)
### Summary of Results
- Center: \((-1, -2)\)
- Foci: \((-6, -2)\) and \((4, -2)\)
- Endpoints of the Latus Rectum: \((-6, 1)\), \((-6, -5)\), \((4, 1)\), \((4, -5)\)
- Length of Major Axis: \(8\)
- Length of Minor Axis: \(6\)
