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What is the center of the hyperbola x2−4y2−100=0x^2 - 4y^2 - 100 = 0?\newline(_,_)(\_,\_)

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Q. What is the center of the hyperbola x2−4y2−100=0x^2 - 4y^2 - 100 = 0?\newline(_,_)(\_,\_)
  1. Rewrite Equation: Rewrite the equation to isolate the constant term on one side.\newlineWe start with the equation x2−4y2−100=0x^2 - 4y^2 - 100 = 0 and move the constant term to the right side to get x2−4y2=100x^2 - 4y^2 = 100.
  2. Standard Form Conversion: Convert the equation into the standard form of a hyperbola.\newlineTo do this, we divide both sides of the equation by 100100 to get (x2)/100−(4y2)/100=1(x^2)/100 - (4y^2)/100 = 1.\newlineSimplifying this, we have x2/100−y2/25=1x^2/100 - y^2/25 = 1.
  3. 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 x2/100−y2/25=1x^2/100 - y^2/25 = 1, we can see that it can be written as (x−0)2/100−(y−0)2/25=1(x - 0)^2/100 - (y - 0)^2/25 = 1. Therefore, h=0h = 0 and k=0k = 0.\newlineCenter of the hyperbola: (0,0)(0, 0).

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