To find the equation of a circle centered at the origin with a given radius, we use the standard form of the equation for a circle.
The standard form of the equation of a circle with center [tex]$(h, k)$[/tex] and radius [tex]$r$[/tex] is:
[tex]$(x - h)^2 + (y - k)^2 = r^2.$[/tex]
In this case, the circle is centered at the origin, so the center [tex]$(h, k)$[/tex] is [tex]$(0, 0)$[/tex]. Therefore, the equation simplifies to:
[tex]$(x - 0)^2 + (y - 0)^2 = r^2.$[/tex]
This further simplifies to:
[tex]$x^2 + y^2 = r^2.$[/tex]
Given that the radius [tex]$r$[/tex] is 4, we substitute [tex]$r = 4$[/tex] into the equation:
[tex]$x^2 + y^2 = 4^2.$[/tex]
Thus, the equation of the circle centered at the origin with a radius of 4 is:
[tex]$x^2 + y^2 = 4^2.$[/tex]
Therefore, the correct answer is:
B. [tex]$x^2 + y^2 = 4^2$[/tex]
