Question
Convert the equation \( \rho=8 \) to rectangular coordinates and write in standard form.
Ask by Harris Lynch. in the United States
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation \( \rho = 8 \) in rectangular coordinates is \( x^2 + y^2 = 64 \).
Solution
To convert the equation \( \rho = 8 \) from polar 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, this distance can be expressed as:
\[
\rho = \sqrt{x^2 + y^2}
\]
Given the equation \( \rho = 8 \), we can substitute this into the equation:
\[
\sqrt{x^2 + y^2} = 8
\]
Next, we will square both sides to eliminate the square root:
\[
x^2 + y^2 = 8^2
\]
Calculating \( 8^2 \):
\[
x^2 + y^2 = 64
\]
This is the equation of a circle in standard form, where the center is at the origin \((0, 0)\) and the radius is \(8\).
Thus, the equation \( \rho = 8 \) in rectangular coordinates and standard form is:
\[
x^2 + y^2 = 64
\]
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Simplify this solution Beyond the Answer
To convert the equation \( \rho = 8 \) from polar to rectangular coordinates, we use the relation \( \rho = \sqrt{x^2 + y^2} \). Substituting this into the equation gives us: \[ \sqrt{x^2 + y^2} = 8 \] Next, we square both sides to eliminate the square root: \[ x^2 + y^2 = 64 \] This is the equation of a circle centered at the origin (0, 0) with a radius of 8. It's already in standard form, which is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. Thus, the final equation in standard form is: \[ x^2 + y^2 = 64 \]
