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Question 1209661: Find the conic section represented by the equation
3x^2 + y^2 + 9x - 5y - 20 = 8x^2 + 6x + 47
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to identify the conic section:
1. **Rearrange the equation:** Move all terms to one side:
3x² + y² + 9x - 5y - 20 - 8x² - 6x - 47 = 0
-5x² + 3x + y² - 5y - 67 = 0
2. **Multiply by -1 (optional, but makes some calculations easier):**
5x² - 3x - y² + 5y + 67 = 0
3. **Complete the square for both x and y:**
5(x² - (3/5)x) - (y² - 5y) + 67 = 0
5(x² - (3/5)x + (3/10)²) - 5(3/10)² - (y² - 5y + (5/2)²) + (5/2)² + 67 = 0
5(x - 3/10)² - 5(9/100) - (y - 5/2)² + 25/4 + 67 = 0
5(x - 3/10)² - 9/20 - (y - 5/2)² + 25/4 + 67 = 0
5(x - 3/10)² - (y - 5/2)² + 67 - 9/20 + 125/20 = 0
5(x - 3/10)² - (y - 5/2)² + 67 + 116/20 = 0
5(x - 3/10)² - (y - 5/2)² + 67 + 5.8 = 0
5(x - 3/10)² - (y - 5/2)² + 72.8 = 0
5(x - 3/10)² - (y - 5/2)² = -72.8
4. **Divide by -72.8:**
[5(x - 3/10)²] / -72.8 - [(y - 5/2)²] / -72.8 = 1
[(x - 3/10)²] / (-72.8/5) - [(y - 5/2)²] / -72.8 = 1
[(x - 3/10)²] / -14.56 - [(y - 5/2)²] / -72.8 = 1
5. **Analyze the equation:**
The equation is in the form:
(x²/a²) - (y²/b²) = 1 (Hyperbola)
OR
-(x²/a²) + (y²/b²) = 1 (Hyperbola)
Since both the x² and y² terms are present and have *opposite* signs, the equation represents a **hyperbola**. The negative signs simply determine the orientation of the hyperbola (whether it opens horizontally or vertically). Since the x^2 term is negative, this is a hyperbola that opens up and down.
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