If for a conic section a focus is (-1,1), eccentricity=3 and the equation of the corresponding directrix is x-y+3=0, then the equation of this conic section is

Options

A. `7x^(2)-18xy+7y^(2)+50x-50y+77=0`
B. `7x^(2)+18xy+7y^(2)=1`
C. `7x^(2)+18xy+7y^(2)-50x+50y+77=0`
D. none

Answer

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

### Step 1: Identify the given values
- Focus (F) = (-1, 1)
- Eccentricity (e) = 3
- Directrix: \( x - y + 3 = 0 \)

### Step 2: Set up the point P
Let P be a point on the conic section with coordinates (x, y).

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

### Step 4: Calculate the distance from P to the directrix
The distance \( PD \) from point P to the directrix can be calculated using the formula for the distance from a point to a line \( Ax + By + C = 0 \):
\[ PD = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
For the directrix \( x - y + 3 = 0 \), we have:
- \( A = 1 \), \( B = -1 \), \( C = 3 \)
Thus,
\[ PD = \frac{|1 \cdot x - 1 \cdot y + 3|}{\sqrt{1^2 + (-1)^2}} = \frac{|x - y + 3|}{\sqrt{2}} \]

### Step 5: Set up the equation based on the definition of conics
According to the definition of a conic section, the distance from the point P to the focus is equal to the eccentricity times the distance from P to the directrix:
\[ PF = e \cdot PD \]
Substituting the expressions we found:
\[ \sqrt{(x + 1)^2 + (y - 1)^2} = 3 \cdot \frac{|x - y + 3|}{\sqrt{2}} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x + 1)^2 + (y - 1)^2 = 9 \cdot \frac{(x - y + 3)^2}{2} \]

### Step 7: Expand both sides
Expanding the left side:
\[ (x + 1)^2 + (y - 1)^2 = x^2 + 2x + 1 + y^2 - 2y + 1 = x^2 + y^2 + 2x - 2y + 2 \]
Expanding the right side:
\[ 9 \cdot \frac{(x - y + 3)^2}{2} = \frac{9}{2} (x^2 - 2xy + y^2 + 6x - 6y + 9) = \frac{9}{2} x^2 - 9xy + \frac{9}{2} y^2 + 27x - 27y + \frac{81}{2} \]

### Step 8: Combine and simplify
Setting both sides equal:
\[ x^2 + y^2 + 2x - 2y + 2 = \frac{9}{2} x^2 - 9xy + \frac{9}{2} y^2 + 27x - 27y + \frac{81}{2} \]
Bringing all terms to one side gives:
\[ 0 = \frac{9}{2} x^2 - x^2 - 9xy + \frac{9}{2} y^2 - y^2 + 27x - 2x - 27y + 2 + \frac{81}{2} \]
This simplifies to:
\[ \frac{7}{2} x^2 + \frac{7}{2} y^2 - 9xy + 25x - 25y + \frac{85}{2} = 0 \]

### Step 9: Multiply through by 2 to eliminate fractions
Multiplying through by 2 gives:
\[ 7x^2 + 7y^2 - 18xy + 50x - 50y + 85 = 0 \]

### Final Equation
Thus, the equation of the conic section is:
\[ 7x^2 + 7y^2 - 18xy + 50x - 50y + 85 = 0 \]