What is the equation of a circle with center at (0,0) and radius r?

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Multiple Choice

What is the equation of a circle with center at (0,0) and radius r?

The equation of a circle in the Cartesian coordinate system can be derived from its definition. A circle is defined as the set of all points that are a fixed distance (the radius) from a central point (the center of the circle). In this case, the center of the circle is located at the origin (0,0), and we denote the radius as r. The general formula for a circle centered at (h, k) with radius r is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] By substituting h = 0 and k = 0 for our center at (0,0), the equation simplifies to: \[ x^2 + y^2 = r^2 \] This indicates that the distance from any point (x, y) on the circle to the center (0, 0) is exactly r, which is the definition of a circle. Thus, the correct equation of the circle with center (0,0) and radius r is \( x^2 + y^2 = r^2 \).

The equation of a circle in the Cartesian coordinate system can be derived from its definition. A circle is defined as the set of all points that are a fixed distance (the radius) from a central point (the center of the circle).

In this case, the center of the circle is located at the origin (0,0), and we denote the radius as r. The general formula for a circle centered at (h, k) with radius r is given by:

[

(x - h)^2 + (y - k)^2 = r^2

]

By substituting h = 0 and k = 0 for our center at (0,0), the equation simplifies to:

[

x^2 + y^2 = r^2

]

This indicates that the distance from any point (x, y) on the circle to the center (0, 0) is exactly r, which is the definition of a circle.

Thus, the correct equation of the circle with center (0,0) and radius r is ( x^2 + y^2 = r^2 ).

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