SOLUTION: Find the conic section represented by the equation
4x^2 - 8y^2 - 28x + 49 = 4x^2 + 5x - 11y + 20
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Question 1209664: Find the conic section represented by the equation
4x^2 - 8y^2 - 28x + 49 = 4x^2 + 5x - 11y + 20
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
To find the conic section represented by the equation 4x² - 8y² - 28x + 49 = 4x² + 5x - 11y + 20, we first need to simplify and rearrange the equation.
1. **Simplify the equation:**
Subtract 4x² from both sides:
-8y² - 28x + 49 = 5x - 11y + 20
Move all terms to one side:
-8y² - 28x - 5x + 11y + 49 - 20 = 0
-8y² - 33x + 11y + 29 = 0
2. **Rearrange the terms:**
-8y² + 11y - 33x + 29 = 0
Now, let's analyze the equation to determine the type of conic section.
* **Check for xy term:** There is no xy term, which simplifies the analysis.
* **Check the coefficients of x² and y²:** The coefficient of x² is 0, and the coefficient of y² is -8. Since one of the squared terms is missing, and the other squared term exists, this indicates that the equation represents a parabola.
To confirm this, we can rewrite the equation in the standard form of a parabola. However, it's easier to observe that since there is a y² term and an x term (but no x² term), the equation represents a parabola.
Therefore, the conic section represented by the equation 4x² - 8y² - 28x + 49 = 4x² + 5x - 11y + 20 is a **parabola**.
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