Convert the equation \( \rho=7 \) to rectangular coordinates and write in standard form.

To convert the equation \( \rho = 7 \) to rectangular coordinates, we start by recognizing that in polar coordinates, \( \rho \) represents the radial distance from the origin. Hence, \( \rho = 7 \) indicates all points that are exactly 7 units away from the origin. In rectangular coordinates, this translates to the equation of a circle. Specifically, the standard form of a circle's equation is given by \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center of the circle and \( r \) is its radius. Here, the center of our circle is at the origin \( (0, 0) \) and the radius \( r \) is 7. Substituting these values into the formula, we get: \[ x^2 + y^2 = 7^2 \] Simplifying this gives us: \[ x^2 + y^2 = 49 \] So the equation \( \rho = 7 \) in rectangular coordinates is \( x^2 + y^2 = 49 \) and this represents a circle with a radius of 7 centered at the origin.