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What is the center of the hyperbola 4x2−y2=1004x^2 - y^2 = 100?\newline(____,____)

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Q. What is the center of the hyperbola 4x2−y2=1004x^2 - y^2 = 100?\newline(____,____)
  1. Write Equation in Form: We start by writing the given equation of the hyperbola in a form that allows us to identify the center directly.\newlineThe given equation is 4x2−y2=1004x^2 - y^2 = 100.
  2. Find Center: To find the center, we need to express the equation in the standard form of a hyperbola. The standard form for a hyperbola that is centered at the point (h,k)(h, k) and opens along the x-axis is (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1. We will rearrange the given equation to match this form.
  3. Normalize Equation: First, we divide both sides of the equation by 100100 to normalize the right side to 11. \newline4x2100−y2100=100100\frac{4x^2}{100} - \frac{y^2}{100} = \frac{100}{100}\newlineSimplifying, we get:\newlinex225−y2100=1\frac{x^2}{25} - \frac{y^2}{100} = 1
  4. Identify Standard Form: Now, we can see that the equation is in the standard form of a hyperbola. The terms (x−h)2/a2(x-h)^2/a^2 and (y−k)2/b2(y-k)^2/b^2 are represented by x2/25x^2/25 and y2/100y^2/100, respectively. Since there are no (x−h)(x-h) or (y−k)(y-k) terms, it means that hh and kk are both 00. Therefore, the center of the hyperbola is at (h,k)=(0,0)(h, k) = (0, 0).

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