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What is the center of the hyperbola 4x2−y2−100=04x^2 - y^2 - 100 = 0?\newline(_,_)(\_,\_)

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Q. What is the center of the hyperbola 4x2−y2−100=04x^2 - y^2 - 100 = 0?\newline(_,_)(\_,\_)
  1. Rewrite equation in standard form: Rewrite the equation of the hyperbola in standard form.\newlineThe given equation is 4x2−y2−100=04x^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)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 for a horizontal hyperbola or (y−k)2a2−(x−h)2b2=1\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 for a vertical hyperbola, where (h,k)(h, k) is the center of the hyperbola.\newlineFirst, we move the constant term to the right side of the equation:\newline4x2−y2=1004x^2 - y^2 = 100
  2. Move constant term to right side: Divide the equation by the constant term on the right to get 11 on the right side.\newlineDivide both sides of the equation by 100100 to get the equation in the form of x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1:\newline4x2100−y2100=100100\frac{4x^2}{100} - \frac{y^2}{100} = \frac{100}{100}\newlineSimplify the fractions:\newlinex225−y2100=1\frac{x^2}{25} - \frac{y^2}{100} = 1
  3. Divide equation by constant term: Identify the center of the hyperbola.\newlineThe equation x225−y2100=1\frac{x^2}{25} - \frac{y^2}{100} = 1 is now in standard form. The center (h,k)(h, k) of the hyperbola is the point (0,0)(0, 0) because there are no terms to shift the hyperbola left/right or up/down from the origin.

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