Find the equation of ellipse having focus at (1,2) corresponding directirx `x-y=2=0` and eccentricity `0.5`.

Explanation

To find the equation of the ellipse with the given parameters, we will follow these steps:

### Step 1: Identify the given parameters
- Focus (S) = (1, 2)
- Directrix (D): x - y - 2 = 0
- Eccentricity (e) = 0.5

### Step 2: Use the definition of the ellipse
According to the definition of an ellipse, for any point P(x, y) on the ellipse, the distance from P to the focus (S) is equal to the eccentricity (e) times the distance from P to the directrix (D).

### Step 3: Calculate the distance from P to the focus
The distance from point P(x, y) to the focus S(1, 2) is given by:
\[ SP = \sqrt{(x - 1)^2 + (y - 2)^2} \]

### Step 4: Calculate the distance from P to the directrix
The distance from point P(x, y) to the directrix line x - y - 2 = 0 can be calculated using the formula for the distance from a point to a line:
\[ PM = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
where A = 1, B = -1, and C = -2. Thus, we have:
\[ PM = \frac{|1 \cdot x - 1 \cdot y - 2|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y - 2|}{\sqrt{2}} \]

### Step 5: Set up the equation using the definition of the ellipse
Using the definition:
\[ SP = e \cdot PM \]
Substituting the distances calculated:
\[ \sqrt{(x - 1)^2 + (y - 2)^2} = 0.5 \cdot \frac{|x - y - 2|}{\sqrt{2}} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 1)^2 + (y - 2)^2 = \left(0.5 \cdot \frac{|x - y - 2|}{\sqrt{2}}\right)^2 \]
This simplifies to:
\[ (x - 1)^2 + (y - 2)^2 = \frac{0.25}{2} (x - y - 2)^2 \]
\[ (x - 1)^2 + (y - 2)^2 = \frac{1}{8} (x - y - 2)^2 \]

### Step 7: Clear the fraction by multiplying through by 8
Multiplying through by 8 gives:
\[ 8((x - 1)^2 + (y - 2)^2) = (x - y - 2)^2 \]

### Step 8: Expand both sides
Expanding both sides:
\[ 8((x^2 - 2x + 1) + (y^2 - 4y + 4)) = (x^2 - 2xy + y^2 - 4x + 4y + 4) \]
This simplifies to:
\[ 8x^2 - 16x + 8 + 8y^2 - 32y + 32 = x^2 - 2xy + y^2 - 4x + 4y + 4 \]

### Step 9: Combine like terms
Rearranging gives:
\[ (8x^2 - x^2) + (8y^2 - y^2) + 2xy - (16x + 4x) - (32y - 4y) + (8 + 32 - 4) = 0 \]
This results in:
\[ 7x^2 + 7y^2 + 2xy - 20x - 28y + 36 = 0 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ 7x^2 + 7y^2 + 2xy - 20x - 28y + 36 = 0 \]