Convert the equation $$ \rho=5 $$ to rectangular coordinates and write in standard form

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To convert the equation $$ \rho = 5 $$ from polar coordinates to rectangular coordinates, we can use the relationship between polar and rectangular coordinates. In polar coordinates, $$ \rho $$ represents the distance from the origin, and in rectangular coordinates, we have:

$$
x = \rho \cos(\theta)
$$
$$
y = \rho \sin(\theta)
$$

The equation $$ \rho = 5 $$ indicates that the distance from the origin is constant at 5. This describes a circle centered at the origin with a radius of 5.

To express this in rectangular coordinates, we can use the relationship:

$$
\rho^2 = x^2 + y^2
$$

Substituting $$ \rho = 5 $$ into this equation gives:

$$
5^2 = x^2 + y^2
$$

This simplifies to:

$$
25 = x^2 + y^2
$$

Now, to write this in standard form, we can rearrange it as:

$$
x^2 + y^2 = 25
$$

This is the standard form of the equation of a circle with center at the origin (0, 0) and radius 5.

Thus, the final answer is:

$$
x^2 + y^2 = 25
$$
The equation $$ \rho = 5 $$ in rectangular coordinates is $$ x^2 + y^2 = 25 $$.

Convert the equation \( \rho=5 \) to rectangular coordinates and write in standard form

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