The directrix of a conic section is the line `3x+4y=1` and the focus S is (-2, 3). If the eccentricity e is `1/sqrt(2)`, find the equation to the conic section.

Answer

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

### Step 1: Identify the given information
- Directrix: \(3x + 4y = 1\)
- Focus \(S\): \((-2, 3)\)
- Eccentricity \(e\): \(\frac{1}{\sqrt{2}}\)

### Step 2: Set up the point \(P\) on the conic
Let \(P\) be a point on the conic with coordinates \((x, y)\). The distance from the focus \(S\) to point \(P\) is denoted as \(SP\), and the distance from point \(P\) to the directrix is denoted as \(Pm\).

### Step 3: Use the definition of eccentricity
According to the definition of eccentricity for a conic section:
\[ \frac{SP}{Pm} = e \]
This can be rearranged to:
\[ SP = e \cdot Pm \]

### Step 4: Calculate \(SP\)
Using the distance formula, the distance \(SP\) from the focus \(S(-2, 3)\) to point \(P(x, y)\) is:
\[ SP = \sqrt{(x + 2)^2 + (y - 3)^2} \]

### Step 5: Calculate \(Pm\)
The distance \(Pm\) from point \(P(x, y)\) to the directrix \(3x + 4y - 1 = 0\) can be calculated using the formula for the distance from a point to a line:
\[ Pm = \frac{|3x + 4y - 1|}{\sqrt{3^2 + 4^2}} = \frac{|3x + 4y - 1|}{5} \]

### Step 6: Substitute into the eccentricity equation
Substituting \(SP\) and \(Pm\) into the eccentricity equation gives:
\[ \sqrt{(x + 2)^2 + (y - 3)^2} = \frac{1}{\sqrt{2}} \cdot \frac{|3x + 4y - 1|}{5} \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (x + 2)^2 + (y - 3)^2 = \frac{1}{2} \cdot \left(\frac{|3x + 4y - 1|}{5}\right)^2 \]
This simplifies to:
\[ (x + 2)^2 + (y - 3)^2 = \frac{(3x + 4y - 1)^2}{50} \]

### Step 8: Expand both sides
Expanding the left side:
\[ (x^2 + 4x + 4) + (y^2 - 6y + 9) = x^2 + y^2 + 4x - 6y + 13 \]

Expanding the right side:
\[ \frac{(3x + 4y - 1)^2}{50} = \frac{9x^2 + 24xy + 16y^2 - 6x - 8y - 2}{50} \]

### Step 9: Multiply through by 50 to eliminate the fraction
Multiplying both sides by 50 gives:
\[ 50(x^2 + y^2 + 4x - 6y + 13) = 9x^2 + 24xy + 16y^2 - 6x - 8y - 2 \]

### Step 10: Rearrange and simplify
Rearranging the equation leads to:
\[ 41x^2 + 34y^2 + 206x - 292y + 199 = 0 \]

### Final Equation
Thus, the equation of the conic section is:
\[ 41x^2 + 34y^2 + 206x - 292y + 199 = 0 \]