To determine the general form of the equation for the circle centered at the origin [tex]\( O(0,0) \)[/tex], we start by recalling the standard equation of a circle centered at [tex]\( (h,k) \)[/tex] with radius [tex]\( r \)[/tex]:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Since the center of the circle is at [tex]\( (0,0) \)[/tex], the equation simplifies to:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Now, we need to match this simplified form with one of the given options:
A. [tex]\( x^2 + y^2 + 41 = 0 \)[/tex]
When this equation is rewritten, it becomes:
[tex]\[ x^2 + y^2 = -41 \][/tex]
Since the right side is negative, it does not represent a real circle. A circle's radius squared [tex]\( (r^2) \)[/tex] must be non-negative.
B. [tex]\( x^2 + y^2 - 41 = 0 \)[/tex]
When this equation is rewritten, it becomes:
[tex]\[ x^2 + y^2 = 41 \][/tex]
This matches the standard form for a circle centered at the origin with [tex]\( r^2 = 41 \)[/tex].
C. [tex]\( x^2 + y^2 + x + y - 41 = 0 \)[/tex]
This equation includes linear terms [tex]\( x \)[/tex] and [tex]\( y \)[/tex], which means it does not fit the standard circle equation that only has [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex] terms.
D. [tex]\( x^2 + y^2 + x - y - 41 = 0 \)[/tex]
Similarly, this equation includes linear terms [tex]\( x \)[/tex] and [tex]\( y \)[/tex], which means it also does not fit the standard circle equation.
Based on the analysis above, the correct form of the equation for a circle centered at [tex]\( O(0,0) \)[/tex] among the given options is:
B. [tex]\( x^2 + y^2 - 41 = 0 \)[/tex]
