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What is the center of the hyperbola 9x2−y2=819x^2 - y^2 = 81?\newline(_,_)(\_,\_)

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Q. What is the center of the hyperbola 9x2−y2=819x^2 - y^2 = 81?\newline(_,_)(\_,\_)
  1. Identify Equation: Identify the equation of the hyperbola.\newlineThe given equation is 9x2−y2=819x^2 - y^2 = 81.
  2. Rewrite Equation: Rewrite the equation in standard form.\newlineTo do this, we need to isolate the terms with variables on one side and the constant on the other side. Since the equation is already in this form, we can proceed to the next step.
  3. Divide by Constant: Divide the equation by the constant term to get the standard form of the hyperbola equation.\newlineDivide both sides of the equation by 8181 to get 9x281−y281=8181\frac{9x^2}{81} - \frac{y^2}{81} = \frac{81}{81}.\newlineSimplify to get x29−y281=1\frac{x^2}{9} - \frac{y^2}{81} = 1.
  4. Identify Center: Identify the center of the 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, where (h,k)(h, k) is the center of the hyperbola. In our equation x2/9−y2/81=1x^2/9 - y^2/81 = 1, we can see that h=0h = 0 and k=0k = 0, so the center is at the origin (0,0)(0, 0).

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