Question 1209666
To find the conic section represented by the equation -x² + 2y² - 8x + 10y - 43 = 3y² - 17x + 14, we first need to simplify and rearrange the equation.

1. **Simplify the equation:**
   Move all terms to one side:
   -x² + 2y² - 8x + 10y - 43 - 3y² + 17x - 14 = 0
   -x² - y² + 9x + 10y - 57 = 0

2. **Rearrange the terms:**
   -x² + 9x - y² + 10y - 57 = 0

3. **Multiply by -1 to make the x² term positive:**
    x² - 9x + y² - 10y + 57 = 0

4. **Complete the square for x and y:**
   (x² - 9x + 81/4) + (y² - 10y + 25) + 57 - 81/4 - 25 = 0
   (x - 9/2)² + (y - 5)² + 57 - 20.25 - 25 = 0
   (x - 9/2)² + (y - 5)² + 11.75 = 0
   (x - 9/2)² + (y - 5)² = -11.75

Since the right side of the equation is negative, this equation does not represent a real conic section. There are no real values of x and y that can satisfy this equation.

Therefore, the given equation does not represent any standard conic section (circle, ellipse, parabola, or hyperbola) in the real plane. It's an *imaginary* conic section.