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Question 1209662: Find the conic section represented by the equation
y^2 + 3x + 11y + 18 = -x^2 - 17y + 59
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! To find the conic section represented by the equation y² + 3x + 11y + 18 = -x² - 17y + 59, we need to simplify and rearrange the equation into a standard form.
1. **Combine like terms and move all terms to one side:**
y² + 3x + 11y + 18 + x² + 17y - 59 = 0
x² + 3x + y² + 28y - 41 = 0
2. **Complete the square for both x and y terms:**
(x² + 3x + (3/2)²) + (y² + 28y + 14²) - 41 - (3/2)² - 14² = 0
(x + 3/2)² + (y + 14)² - 41 - 9/4 - 196 = 0
(x + 3/2)² + (y + 14)² - (164 + 9 + 784)/4 = 0
(x + 3/2)² + (y + 14)² - 957/4 = 0
(x + 3/2)² + (y + 14)² = 957/4
3. **Analyze the equation:**
The equation is now in the standard form of a circle:
(x - h)² + (y - k)² = r²
In our case, the equation is:
(x + 3/2)² + (y + 14)² = 957/4
This is the equation of a circle with center (-3/2, -14) and radius √(957/4) = √957 / 2.
Therefore, the conic section represented by the given equation is a **circle**.
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