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What is the center of the hyperbola 16x2−y2=6416x^2 - y^2 = 64?\newline(_,_)(\_,\_)

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Q. What is the center of the hyperbola 16x2−y2=6416x^2 - y^2 = 64?\newline(_,_)(\_,\_)
  1. Write Equation: Write the given equation of the hyperbola.\newlineThe given equation is 16x2−y2=6416x^2 - y^2 = 64.
  2. Rearrange Equation: Rearrange the equation to resemble the standard form of a hyperbola.\newlineTo do this, we need to isolate the terms with xx and yy on one side and move the constant to the other side. We have:\newline16x2−y2−64=016x^2 - y^2 - 64 = 0\newlineAdding 6464 to both sides gives us:\newline16x2−y2=6416x^2 - y^2 = 64
  3. Divide Equation: Divide the equation by 6464 to get the standard form of the hyperbola.\newlineDividing each term by 6464, we get:\newline16x264−y264=6464 \frac{16x^2}{64} - \frac{y^2}{64} = \frac{64}{64} \newlineSimplifying this, we have:\newlinex2/4−y2/64=1 x^2/4 - y^2/64 = 1
  4. Identify Center: Identify the center of the hyperbola.\newlineThe standard form of a 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. In our equation x2/4−y2/64=1x^2/4 - y^2/64 = 1, we can see that h=0h = 0 and k=0k = 0, since there are no terms to shift the hyperbola left/right or up/down.\newlineTherefore, the center of the hyperbola is (0,0)(0, 0).

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