Find the equation of the ellipse whose co-ordinate of focus are (6,7), equation of directrix x+y+1= and eccentricity is `(1)/(sqrt(2))`.

Answer

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

### Step 1: Identify the given values
- Focus (F) = (6, 7)
- Directrix: \( x + y + 1 = 0 \)
- Eccentricity (e) = \( \frac{1}{\sqrt{2}} \)

### Step 2: Set up the relationship between the distances
For any point \( P(x, y) \) on the ellipse, the distance from the point to the focus (denoted as \( PO \)) and the distance from the point to the directrix (denoted as \( PN \)) must satisfy the relationship:
\[ PO = e \cdot PN \]

### Step 3: Calculate \( PO \)
Using the distance formula, the distance from point \( P(x, y) \) to the focus \( F(6, 7) \) is:
\[ PO = \sqrt{(x - 6)^2 + (y - 7)^2} \]

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

### Step 5: Substitute into the relationship
Substituting \( PO \) and \( PN \) into the relationship gives:
\[ \sqrt{(x - 6)^2 + (y - 7)^2} = \frac{1}{\sqrt{2}} \cdot \frac{|x + y + 1|}{\sqrt{2}} \]
This simplifies to:
\[ \sqrt{(x - 6)^2 + (y - 7)^2} = \frac{|x + y + 1|}{2} \]

### Step 6: Square both sides
Squaring both sides results in:
\[ (x - 6)^2 + (y - 7)^2 = \left(\frac{|x + y + 1|}{2}\right)^2 \]
This simplifies to:
\[ (x - 6)^2 + (y - 7)^2 = \frac{(x + y + 1)^2}{4} \]

### Step 7: Expand both sides
Expanding the left side:
\[ (x - 6)^2 + (y - 7)^2 = (x^2 - 12x + 36) + (y^2 - 14y + 49) = x^2 + y^2 - 12x - 14y + 85 \]

Expanding the right side:
\[ \frac{(x + y + 1)^2}{4} = \frac{x^2 + 2xy + y^2 + 2x + 2y + 1}{4} \]

### Step 8: Multiply both sides by 4
To eliminate the fraction, multiply both sides by 4:
\[ 4(x^2 + y^2 - 12x - 14y + 85) = x^2 + 2xy + y^2 + 2x + 2y + 1 \]

### Step 9: Rearrange the equation
Rearranging gives:
\[ 4x^2 + 4y^2 - 48x - 56y + 340 = x^2 + 2xy + y^2 + 2x + 2y + 1 \]
Combining like terms results in:
\[ 3x^2 + 3y^2 - 2xy - 50x - 58y + 339 = 0 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ 3x^2 + 3y^2 - 2xy - 50x - 58y + 339 = 0 \]

---