Find the centre foci and the equation of the directrices of the ellipse `8x^(2) +9y^(2) - 16x + 18 y - 55 = 0 `

Answer

To find the center, foci, and the equations of the directrices of the ellipse given by the equation \(8x^2 + 9y^2 - 16x + 18y - 55 = 0\), we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the given equation:
\[ 8x^2 + 9y^2 - 16x + 18y - 55 = 0 \]
This can be rearranged as:
\[ 8x^2 - 16x + 9y^2 + 18y - 55 = 0 \]

### Step 2: Completing the Square
Now, we will complete the square for the \(x\) and \(y\) terms.

**For \(x\):**
\[ 8(x^2 - 2x) = 8((x - 1)^2 - 1) = 8(x - 1)^2 - 8 \]

**For \(y\):**
\[ 9(y^2 + 2y) = 9((y + 1)^2 - 1) = 9(y + 1)^2 - 9 \]

Substituting these back into the equation gives:
\[ 8((x - 1)^2 - 1) + 9((y + 1)^2 - 1) - 55 = 0 \]
This simplifies to:
\[ 8(x - 1)^2 + 9(y + 1)^2 - 8 - 9 - 55 = 0 \]
\[ 8(x - 1)^2 + 9(y + 1)^2 - 72 = 0 \]
\[ 8(x - 1)^2 + 9(y + 1)^2 = 72 \]

### Step 3: Dividing by 72
Now, divide the entire equation by 72 to get it in standard form:
\[ \frac{8(x - 1)^2}{72} + \frac{9(y + 1)^2}{72} = 1 \]
This simplifies to:
\[ \frac{(x - 1)^2}{9} + \frac{(y + 1)^2}{8} = 1 \]

### Step 4: Identifying Parameters
From the standard form \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), we identify:
- Center \((h, k) = (1, -1)\)
- \(a^2 = 9 \Rightarrow a = 3\)
- \(b^2 = 8 \Rightarrow b = 2\sqrt{2}\)

### Step 5: Finding the Foci
The distance of the foci from the center is given by:
\[ c = \sqrt{a^2 - b^2} = \sqrt{9 - 8} = \sqrt{1} = 1 \]
Thus, the foci are located at:
\[ (h \pm c, k) = (1 \pm 1, -1) = (2, -1) \text{ and } (0, -1) \]

### Step 6: Finding the Directrices
The equations of the directrices are given by:
\[ x = h \pm \frac{a}{e} \]
where \(e = \frac{c}{a} = \frac{1}{3}\).

Calculating the directrices:
\[ \frac{a}{e} = \frac{3}{\frac{1}{3}} = 9 \]
Thus, the equations of the directrices are:
\[ x = 1 \pm 9 \Rightarrow x = 10 \text{ and } x = -8 \]

### Summary of Results
- **Center:** \((1, -1)\)
- **Foci:** \((2, -1)\) and \((0, -1)\)
- **Directrices:** \(x = 10\) and \(x = -8\)