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Question 1209666: Find the conic section represented by the equation
-x^2 + 2y^2 - 8x + 10y - 43 = 3y^2 - 17x + 14
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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