SOLUTION: Find the conic section represented by the equation 5x^2 + y^2 + 10x - 4y + 17 = -4y^2

SOLUTION: Find the conic section represented by the equation 5x^2 + y^2 + 10x - 4y + 17 = -4y^2 - 18x + 12y

Question 1209663: Find the conic section represented by the equation
5x^2 + y^2 + 10x - 4y + 17 = -4y^2 - 18x + 12y
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
To determine the conic section represented by the equation 5x² + y² + 10x - 4y + 17 = -4y² - 18x + 12y, we need to simplify and rearrange the equation into a standard form.
1. **Combine like terms and move all terms to one side:**
5x² + y² + 10x - 4y + 17 + 4y² + 18x - 12y = 0
5x² + 28x + 5y² - 16y + 17 = 0
2. **Complete the square for both x and y terms:**
5(x² + (28/5)x) + 5(y² - (16/5)y) + 17 = 0
5(x² + (28/5)x + (14/5)²) - 5(14/5)² + 5(y² - (16/5)y + (8/5)²) - 5(8/5)² + 17 = 0
5(x + 14/5)² - 5(196/25) + 5(y - 8/5)² - 5(64/25) + 17 = 0
5(x + 14/5)² - 196/5 + 5(y - 8/5)² - 64/5 + 85/5 = 0
5(x + 14/5)² + 5(y - 8/5)² - 175/5 = 0
5(x + 14/5)² + 5(y - 8/5)² = 35
(x + 14/5)² + (y - 8/5)² = 7
3. **Analyze the equation:**
The equation is now in the standard form of an ellipse:
(x - h)²/a² + (y - k)²/b² = 1
In our case, the equation is:
(x + 14/5)²/7 + (y - 8/5)²/7 = 1
This is the equation of a circle (which is a special case of an ellipse where a = b) with center (-14/5, 8/5) and radius √7.
Therefore, the conic section represented by the given equation is a **circle**.

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