To determine the correct equation of a circle centered at the origin [tex]\(O(0,0)\)[/tex], we need to analyze the general form of the equation of a circle. The equation of a circle centered at the origin with radius [tex]\(r\)[/tex] is given by:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Let's examine each given option to determine if it fits this form:
Option A: [tex]\(x^2 + y^2 + 41 = 0\)[/tex]
- Rearrange this equation into the standard form:
[tex]\[ x^2 + y^2 = -41 \][/tex]
- This implies that [tex]\(r^2 = -41\)[/tex]. Since the radius squared ([tex]\(r^2\)[/tex]) cannot be negative, this equation is not possible.
Option B: [tex]\(x^2 + y^2 - 41 = 0\)[/tex]
- Rearrange this equation into the standard form:
[tex]\[ x^2 + y^2 = 41 \][/tex]
- This implies that [tex]\(r^2 = 41\)[/tex]. The radius squared is a positive value, which is possible. Thus, this option represents a circle centered at the origin with a radius [tex]\( \sqrt{41} \)[/tex].
Option C: [tex]\(x^2 + y^2 + x + y - 41 = 0\)[/tex]
- This equation contains the terms [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. The presence of these linear terms ([tex]\(+x+y\)[/tex]) indicates that the circle is not centered at the origin but is instead shifted. Therefore, this does not match the general form of a circle centered at [tex]\( (0,0) \)[/tex] where we only have [tex]\(x^2 + y^2\)[/tex] terms.
Option D: [tex]\(x^2 + y^2 + x - y - 41 = 0\)[/tex]
- Similar to Option C, this equation also contains linear terms ([tex]\(+x\)[/tex] and [tex]\(-y\)[/tex]), indicating the center of the circle is not at the origin. Thus, this does not match our criteria for a circle centered at [tex]\( (0,0) \)[/tex].
Based on the analysis, only Option B fits the equation of a circle centered at the origin:
[tex]\[ x^2 + y^2 - 41 = 0 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
