The equation of conic section whose focus is at (-1 , 0), directrix is the 4x-3y+2=0 and eccentricity `1//sqrt2`1, is

Options

A. `34x^(2)+41y^(2)+24xy+84x+12y+46=0`
B. `34x^(2)+41y^(2)-24xy+84x+12y+46=0`
C. `34x^(2)+41y^(2)-24xy-84x-12y+46=0`
D. none of these.

Explanation

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

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Focus \( S = (-1, 0) \)
- Directrix \( 4x - 3y + 2 = 0 \)
- Eccentricity \( e = \frac{1}{\sqrt{2}} \)

2. **Let the Point on the Conic Section be \( P(h, k) \)**:
- We will use the definition of a conic section, which states that the distance from the focus to a point \( P \) is equal to the eccentricity times the perpendicular distance from the point \( P \) to the directrix.

3. **Calculate the Distance from the Focus to the Point \( P \)**:
- The distance \( SP \) from the focus \( S(-1, 0) \) to the point \( P(h, k) \) is given by:
\[ SP = \sqrt{(h + 1)^2 + k^2} \]

4. **Calculate the Perpendicular Distance from the Point \( P \) to the Directrix**:
- The formula for the distance from a point \( (h, k) \) to the line \( Ax + By + C = 0 \) is:
\[ \text{Distance} = \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \]
- For the line \( 4x - 3y + 2 = 0 \), we have \( A = 4, B = -3, C = 2 \). Thus, the distance \( PM \) is:
\[ PM = \frac{|4h - 3k + 2|}{\sqrt{4^2 + (-3)^2}} = \frac{|4h - 3k + 2|}{5} \]

5. **Set Up the Equation Using the Definition of Conic Section**:
- According to the definition:
\[ SP = e \cdot PM \]
- Substituting the distances:
\[ \sqrt{(h + 1)^2 + k^2} = \frac{1}{\sqrt{2}} \cdot \frac{|4h - 3k + 2|}{5} \]

6. **Square Both Sides to Eliminate the Square Root**:
- Squaring both sides gives:
\[ (h + 1)^2 + k^2 = \frac{1}{2} \cdot \left(\frac{|4h - 3k + 2|}{5}\right)^2 \]
- This simplifies to:
\[ (h + 1)^2 + k^2 = \frac{(4h - 3k + 2)^2}{50} \]

7. **Expand Both Sides**:
- Left side:
\[ (h + 1)^2 + k^2 = h^2 + 2h + 1 + k^2 \]
- Right side:
\[ \frac{(4h - 3k + 2)^2}{50} = \frac{16h^2 - 24hk + 9k^2 + 16h - 12k + 4}{50} \]

8. **Multiply Through by 50 to Eliminate the Denominator**:
- This gives:
\[ 50(h^2 + 2h + 1 + k^2) = 16h^2 - 24hk + 9k^2 + 16h - 12k + 4 \]

9. **Rearrange the Equation**:
- Combine like terms:
\[ 34h^2 + 41k^2 + 84h + 24hk + 46 = 0 \]

10. **Replace \( h \) and \( k \) with \( x \) and \( y \)**:
- The final equation of the conic section is:
\[ 34x^2 + 41y^2 + 84x + 24xy + 46 = 0 \]

### Final Answer:
The equation of the conic section is:
\[ 34x^2 + 41y^2 + 84x + 24xy + 46 = 0 \]