The equation of the hyperbola whose focus is `(1,2)`, directrix is the line `x+y+1=0` and eccentricity `3//2`, is

Options

A. `x^(2)+y^(2)+18xy+34x+50y-31=0`
B. `x^(2)+y^(2)-18xy-2x-14y+31=0`
C. `x^(2)+y^(2)+18xy+34x+50y+49=0`
D. `x^(2)+y^(2)-18xy+34x+50y-31=0`

Explanation

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

### Step 1: Identify the Given Information
- Focus (F): \( (1, 2) \)
- Directrix: \( x + y + 1 = 0 \)
- Eccentricity (e): \( \frac{3}{2} \)

### Step 2: Determine the Distance from a Point to the Focus
Let \( P(x, y) \) be any point on the hyperbola. The distance from point \( P \) to the focus \( F(1, 2) \) is given by:
\[ PF = \sqrt{(x - 1)^2 + (y - 2)^2} \]

### Step 3: Calculate the Perpendicular Distance from Point to the Directrix
The perpendicular distance \( SP \) from point \( P(x, y) \) to the directrix \( x + y + 1 = 0 \) can be calculated using the formula for the distance from a point to a line:
\[ SP = \frac{|x + y + 1|}{\sqrt{1^2 + 1^2}} = \frac{|x + y + 1|}{\sqrt{2}} \]

### Step 4: Set Up the Hyperbola Definition
According to the definition of a hyperbola, the distance from the focus to the point \( P \) is equal to the eccentricity times the distance from the point \( P \) to the directrix:
\[ PF = e \cdot SP \]
Substituting the expressions we found:
\[ \sqrt{(x - 1)^2 + (y - 2)^2} = \frac{3}{2} \cdot \frac{|x + y + 1|}{\sqrt{2}} \]

### Step 5: Square Both Sides to Eliminate the Square Root
Squaring both sides gives:
\[ (x - 1)^2 + (y - 2)^2 = \left(\frac{3}{2} \cdot \frac{|x + y + 1|}{\sqrt{2}}\right)^2 \]
This simplifies to:
\[ (x - 1)^2 + (y - 2)^2 = \frac{9}{8} (x + y + 1)^2 \]

### Step 6: Expand Both Sides
Expanding the left side:
\[ (x - 1)^2 + (y - 2)^2 = x^2 - 2x + 1 + y^2 - 4y + 4 = x^2 + y^2 - 2x - 4y + 5 \]
Expanding the right side:
\[ \frac{9}{8} (x + y + 1)^2 = \frac{9}{8} (x^2 + 2xy + y^2 + 2x + 2y + 1) \]
This becomes:
\[ \frac{9}{8} x^2 + \frac{9}{8} y^2 + \frac{9}{4} xy + \frac{9}{4} x + \frac{9}{4} y + \frac{9}{8} \]

### Step 7: Rearranging the Equation
Setting both sides equal:
\[ x^2 + y^2 - 2x - 4y + 5 = \frac{9}{8} x^2 + \frac{9}{8} y^2 + \frac{9}{4} xy + \frac{9}{4} x + \frac{9}{4} y + \frac{9}{8} \]
Rearranging gives:
\[ 0 = \left(1 - \frac{9}{8}\right)x^2 + \left(1 - \frac{9}{8}\right)y^2 - 2x - 4y + 5 - \frac{9}{4} xy - \frac{9}{4} x - \frac{9}{4} y - \frac{9}{8} \]

### Step 8: Combine Like Terms
Combining like terms leads to:
\[ -\frac{1}{8}x^2 - \frac{1}{8}y^2 - \frac{9}{4}xy + \left(-2 - \frac{9}{4}\right)x + \left(-4 - \frac{9}{4}\right)y + \left(5 - \frac{9}{8}\right) = 0 \]

### Step 9: Multiply Through by -8 to Clear Fractions
Multiplying through by -8 gives:
\[ x^2 + y^2 + 18xy + 34x + 32y - 40 = 0 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ x^2 + y^2 + 18xy + 34x + 32y - 40 = 0 \]