To find the correct equation representing a circle centered at the origin with a radius of 7, let's understand the general equation for a circle in a coordinate plane.
The standard equation of a circle centered at the origin [tex]\((0,0)\)[/tex] with radius [tex]\(r\)[/tex] is given by:
[tex]\[
x^2 + y^2 = r^2
\][/tex]
In this case, the radius ([tex]\(r\)[/tex]) is given as 7. Plugging the value of the radius into the equation, we get:
[tex]\[
x^2 + y^2 = 7^2
\][/tex]
This simplifies to:
[tex]\[
x^2 + y^2 = 49
\][/tex]
So, the equation of the circle centered at the origin with radius 7 is [tex]\(x^2 + y^2 = 49\)[/tex]. Let's analyze each provided option:
A. [tex]\( x^2 + y^2 = 7^2 \)[/tex]
This option is correct because it correctly represents [tex]\(x^2 + y^2 = 49\)[/tex], as 7^2 is equal to 49.
B. [tex]\( x^2 + y^2 = 7 \)[/tex]
This is not correct, as it implies a radius of [tex]\(\sqrt{7}\)[/tex], not 7.
C. [tex]\( (x - 7)^2 + (y - 7)^2 = 7^2 \)[/tex]
This is not correct, as it represents a circle centered at [tex]\((7, 7)\)[/tex] with radius 7, not at the origin.
D. [tex]\( (x - 7)^2 + y^2 = 49 \)[/tex]
This is not correct either, as it represents a circle centered at [tex]\((7, 0)\)[/tex] with radius 7.
Therefore, the correct option is:
A. [tex]\( x^2 + y^2 = 7^2 \)[/tex]
