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What is the center of the hyperbola x2x^2 - y2y^2 = 100100?\newline(_,_)(\_,\_)

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Q. What is the center of the hyperbola x2x^2 - y2y^2 = 100100?\newline(_,_)(\_,\_)
  1. Rewrite Equation in Standard Form: The equation of the hyperbola is given by x2−y2=100x^2 - y^2 = 100. To find the center, we need to express the equation in its standard form.
  2. Standard Form of Hyperbola: The 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 for a horizontal hyperbola, or (y−k)2/a2−(x−h)2/b2=1(y - k)^2/a^2 - (x - h)^2/b^2 = 1 for a vertical hyperbola, where (h,k)(h, k) is the center of the hyperbola.
  3. Divide by 100100: The given equation x2−y2=100x^2 - y^2 = 100 can be rewritten as x2100−y2100=1\frac{x^2}{100} - \frac{y^2}{100} = 1 by dividing both sides by 100100.
  4. Identify Center Coordinates: Now, the equation x2100−y2100=1\frac{x^2}{100} - \frac{y^2}{100} = 1 is in the standard form of a hyperbola, where a2=100a^2 = 100 and b2=100b^2 = 100. Since there is no (x−h)(x - h) or (y−k)(y - k) term, it implies that h=0h = 0 and k=0k = 0.
  5. Identify Center Coordinates: Now, the equation x2100−y2100=1\frac{x^2}{100} - \frac{y^2}{100} = 1 is in the standard form of a hyperbola, where a2=100a^2 = 100 and b2=100b^2 = 100. Since there is no (x−h)(x - h) or (y−k)(y - k) term, it implies that h=0h = 0 and k=0k = 0.Therefore, the center of the hyperbola is at the point (h,k)(h, k), which is (0,0)(0, 0).

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