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What is the center of the hyperbola x2−4y2=64x^2 - 4y^2 = 64?\newline(_,_(\_,\_)

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Q. What is the center of the hyperbola x2−4y2=64x^2 - 4y^2 = 64?\newline(_,_(\_,\_)
  1. Write Equation: Write the given equation of the hyperbola.\newlineThe given equation is x2−4y2=64x^2 - 4y^2 = 64.
  2. Rearrange for Center: Rearrange the equation to identify the center.\newlineTo find the center, we need to express the equation in the standard form of a hyperbola. The standard form for a horizontal hyperbola is (x−h)2/a2−(y−k)2/b2=1(x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k)(h, k) is the center of the hyperbola. For a vertical hyperbola, the terms are switched.
  3. Divide by 6464: Divide the equation by 6464 to get the standard form.\newlineDivide both sides of the equation by 6464 to get x264−4y264=1\frac{x^2}{64} - \frac{4y^2}{64} = 1.\newlineSimplify to get x264−y216=1\frac{x^2}{64} - \frac{y^2}{16} = 1.
  4. Identify Center: Identify the center of the hyperbola.\newlineFrom the standard form x264−y216=1\frac{x^2}{64} - \frac{y^2}{16} = 1, we can see that the equation can be written as (x−0)2/64−(y−0)2/16=1\left(x-0\right)^2/64 - \left(y-0\right)^2/16 = 1. Therefore, the center (h,k)(h, k) of the hyperbola is (0,0)(0, 0).

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