For the hyperbola x^2/25 - y^2/16 = 1, what is the distance between the foci?

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Q: For the hyperbola x^2/25 - y^2/16 = 1, what is the distance between the foci?

Solution: The distance between the foci of the hyperbola is 2c, where c = √(a^2 + b^2) = √(25 + 16) = √41, so the distance is 2√41.

Steps: 9

Step 1: Identify the standard form of the hyperbola equation, which is x^2/a^2 - y^2/b^2 = 1.
Step 2: From the given equation x^2/25 - y^2/16 = 1, identify a^2 and b^2. Here, a^2 = 25 and b^2 = 16.
Step 3: Calculate a by taking the square root of a^2. So, a = √25 = 5.
Step 4: Calculate b by taking the square root of b^2. So, b = √16 = 4.
Step 5: Use the formula for c, which is c = √(a^2 + b^2). Substitute the values: c = √(25 + 16).
Step 6: Calculate the sum inside the square root: 25 + 16 = 41.
Step 7: Now, find c by calculating √41.
Step 8: The distance between the foci of the hyperbola is given by the formula 2c. So, calculate 2 * √41.
Step 9: The final answer for the distance between the foci is 2√41.