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What is the center of the hyperbola x2−y2=49x^2 - y^2 = 49?\newline(_,_)(\_,\_)

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Find properties of hyperbolas from equations in general formright chevron icon

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Q. What is the center of the hyperbola x2−y2=49x^2 - y^2 = 49?\newline(_,_)(\_,\_)
  1. Identify Standard Form: Identify the standard form of the hyperbola equation.\newlineThe standard form of a hyperbola centered at (h,k)(h, k) 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)2b2−(x−h)2a2=1\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1 for a vertical hyperbola. We need to compare the given equation x2−y2=49x^2 - y^2 = 49 with the standard form to find the center.
  2. Rewrite Equation: Rewrite the given equation in standard form.\newlineThe given equation is x2−y2=49x^2 - y^2 = 49. To compare it with the standard form, we can rewrite it as (x2)/49−(y2)/49=1(x^2)/49 - (y^2)/49 = 1, which simplifies to (x2)/72−(y2)/72=1(x^2)/7^2 - (y^2)/7^2 = 1.
  3. Identify Center: Identify the center of the hyperbola.\newlineFrom the standard form (x272−y272=1(\frac{x^2}{7^2} - \frac{y^2}{7^2} = 1, we can see that h=0h = 0 and k=0k = 0, since there are no terms (x−h)(x - h) or (y−k)(y - k) in the equation. Therefore, the center of the hyperbola is at the origin (0,0)(0, 0).

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