How to find the center of a hyperbola?
Understand the Problem
The question is asking for the method to determine the center of a hyperbola. To solve this, we'll need to refer to the standard equations of hyperbolas, which provide the coordinates for the center based on their mathematical structure.
Answer
The center of the hyperbola is at the coordinates $(3, -2)$.
Answer for screen readers
The center of the hyperbola is at the coordinates $(3, -2)$.
Steps to Solve
- Identify the Standard Form of the Hyperbola
The center of a hyperbola can be determined from its standard form equations:
- For a hyperbola that opens horizontally: $$ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 $$
- For a hyperbola that opens vertically: $$ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 $$
Here, $(h, k)$ represents the center of the hyperbola.
- Determine the Values of (h) and (k)
From the equations above, you can directly find the values of (h) and (k).
- For both equations, (h) and (k) are the numbers that you subtract from (x) and (y) respectively.
- Example Hyperbola
Consider the equation of a hyperbola: $$ \frac{(x-3)^2}{16} - \frac{(y+2)^2}{25} = 1 $$
From this equation, we can identify:
- (h = 3)
- (k = -2)
- State the Center
Thus, the center of the hyperbola is at the point ((h, k)).
The center of the hyperbola is at the coordinates $(3, -2)$.
More Information
The center of a hyperbola is a key point that helps in identifying its orientation and properties. Understanding how to derive it from the equation can assist in graphing the hyperbola efficiently.
Tips
- A common mistake is misidentifying the signs of (h) and (k) when reading them from the equation. Always pay attention to the signs in the equations (for example, (y-k) indicates (k) is subtracted, which could lead to confusion with its sign).
- Not recognizing whether the hyperbola is horizontal or vertical based on the form of the equation can lead to errors in analyzing the graph.
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