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What is the center of the hyperbola x2−y2−100=0x^2 - y^2 - 100 = 0?\newline(_,_)(\_,\_)

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Q. What is the center of the hyperbola x2−y2−100=0x^2 - y^2 - 100 = 0?\newline(_,_)(\_,\_)
  1. Rewrite Equation: Rewrite the given hyperbola equation in standard form.\newlineThe given equation is x2−y2−100=0x^2 - y^2 - 100 = 0. To find the center, we need to express the equation in the standard form of a hyperbola, which 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.\newlineFirst, we move the constant term to the right side of the equation:\newlinex2−y2=100x^2 - y^2 = 100
  2. Divide by Constant: Divide the equation by the constant term on the right side to get the standard form.\newlineDividing both sides by 100100, we get:\newlinex2100−y2100=1\frac{x^2}{100} - \frac{y^2}{100} = 1\newlineThis can be rewritten as:\newlinex2102−y2102=1\frac{x^2}{10^2} - \frac{y^2}{10^2} = 1
  3. Identify Center: Identify the center of the hyperbola. The equation is now in the form (x2102)−(y2102)=1(\frac{x^2}{10^2}) - (\frac{y^2}{10^2}) = 1, which is similar to the standard form ((x−h)2a2)−((y−k)2b2)=1(\frac{(x-h)^2}{a^2}) - (\frac{(y-k)^2}{b^2}) = 1. Since there are no terms (x−h)(x-h) or (y−k)(y-k), we can conclude that h=0h = 0 and k=0k = 0. Therefore, the center of the hyperbola is at the point (h,k)=(0,0)(h, k) = (0, 0).

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