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What is the center of the hyperbola 9x2−y2−81=09x^2 - y^2 - 81 = 0?\newline(_,_)(\_,\_)

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Q. What is the center of the hyperbola 9x2−y2−81=09x^2 - y^2 - 81 = 0?\newline(_,_)(\_,\_)
  1. Write Equation: Write down the given equation of the hyperbola.\newlineThe given equation is 9x2−y2−81=09x^2 - y^2 - 81 = 0.
  2. Move Constant: Move the constant term to the right side of the equation.\newline9x2−y2=819x^2 - y^2 = 81
  3. Divide by 8181: Divide both sides of the equation by 8181 to get the equation in standard form.\newline(9x2)/81−(y2)/81=81/81(9x^2)/81 - (y^2)/81 = 81/81\newlineSimplify the equation to get:\newlinex2/9−y2/81=1x^2/9 - y^2/81 = 1
  4. Simplify Equation: Identify the center of the hyperbola.\newlineThe standard form of the equation 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.\newlineIn our equation x2/9−y2/81=1x^2/9 - y^2/81 = 1, we can see that it matches the form (x−h)2/a2−(y−k)2/b2=1(x-h)^2/a^2 - (y-k)^2/b^2 = 1 with h=0h = 0 and k=0k = 0.\newlineTherefore, the center of the hyperbola is at (0,0)(0, 0).

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