Find the equation of the ellipse with
focus at (1, -1), directrix x = 0, and `e=sqrt(2)/2`,

Answer

To find the equation of the ellipse with the given focus, directrix, and eccentricity, we will follow these steps:

### Step 1: Identify the Given Information
- Focus (F) = (1, -1)
- Directrix (D) = x = 0 (which is the y-axis)
- Eccentricity (e) = √2 / 2

### Step 2: Define a Point on the Ellipse
Let P(h, k) be any point on the ellipse.

### Step 3: Calculate the Distance from Point P to the Focus
Using the distance formula, the distance (FP) from point P(h, k) to the focus F(1, -1) is given by:
\[ FP = \sqrt{(h - 1)^2 + (k + 1)^2} \]

### Step 4: Calculate the Distance from Point P to the Directrix
The distance (DP) from point P(h, k) to the directrix x = 0 is simply the x-coordinate of point P:
\[ DP = |h| \]

### Step 5: Use the Definition of Eccentricity
According to the definition of eccentricity for conic sections:
\[ e = \frac{FP}{DP} \]
Substituting the distances we found:
\[ \frac{\sqrt{(h - 1)^2 + (k + 1)^2}}{|h|} = \frac{\sqrt{2}}{2} \]

### Step 6: Cross-Multiply and Square Both Sides
Cross-multiplying gives:
\[ \sqrt{(h - 1)^2 + (k + 1)^2} = \frac{\sqrt{2}}{2} |h| \]
Squaring both sides results in:
\[ (h - 1)^2 + (k + 1)^2 = \frac{2}{4} h^2 \]
This simplifies to:
\[ (h - 1)^2 + (k + 1)^2 = \frac{1}{2} h^2 \]

### Step 7: Expand and Rearrange the Equation
Expanding the left-hand side:
\[ (h^2 - 2h + 1) + (k^2 + 2k + 1) = \frac{1}{2} h^2 \]
Combining terms gives:
\[ h^2 + k^2 - 2h + 2k + 2 = \frac{1}{2} h^2 \]
Rearranging leads to:
\[ h^2 - \frac{1}{2} h^2 + k^2 - 2h + 2k + 2 = 0 \]
This simplifies to:
\[ \frac{1}{2} h^2 + k^2 - 2h + 2k + 2 = 0 \]

### Step 8: Multiply Through by 2 to Eliminate the Fraction
Multiplying the entire equation by 2 gives:
\[ h^2 + 2k^2 - 4h + 4k + 4 = 0 \]

### Step 9: Replace h and k with x and y
Finally, replacing h with x and k with y, we get the equation of the ellipse:
\[ x^2 + 2y^2 - 4x + 4y + 4 = 0 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ x^2 + 2y^2 - 4x + 4y + 4 = 0 \]

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