Final answer:
The center of a hyperbola is not at the origin if its equation in standard form includes a constant term that is not zero, indicating a horizontal and/or vertical shift from the origin.
Step-by-step explanation:
To determine if the center of a hyperbola is not at the origin, you should look at the form of its equation. If the equation is in its standard form, the presence of a constant term on one side would indicate that the center is not at the origin. Option C is correct here because the equation must have a constant term that is not zero. This term would typically be seen as a subtraction or addition after the x² and y² terms, shifting the hyperbola horizontally and/or vertically on the coordinate plane. The constant term will affect the center's location - if it exists, the center has been moved from the origin.
Options A and B are incorrect because having unequal coefficients or lacking coefficients for x and y do not directly indicate the center's location, as they relate to the shape and orientation of the hyperbola rather than its position on the coordinate plane. Option D is partially correct in stating that the hyperbola must be in standard form, but the non-zero constant is the key to identifying the center's shift from the origin.