Question
To convert the polar equation \( r=4 \) to ectangular form, we use the relationship oetween polar and rectangular coordinates:
Ask by Pena Savage. in Iraq
Jan 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The polar equation \( r = 4 \) converts to the rectangular form \( x^2 + y^2 = 16 \), which represents a circle with a radius of 4 units centered at the origin.
Solution
To convert the polar equation \( r = 4 \) to its rectangular (Cartesian) form, we use the relationships between polar and rectangular coordinates:
\[
\begin{align*}
x &= r \cos \theta, \\
y &= r \sin \theta, \\
r^2 &= x^2 + y^2.
\end{align*}
\]
Here's the step-by-step conversion:
1. **Start with the given polar equation:**
\[
r = 4.
\]
2. **Square both sides to utilize the relationship \( r^2 = x^2 + y^2 \):**
\[
r^2 = 16.
\]
3. **Substitute \( r^2 \) with \( x^2 + y^2 \):**
\[
x^2 + y^2 = 16.
\]
**Final Rectangular Form:**
\[
x^2 + y^2 = 16.
\]
**Interpretation:**
This equation represents a circle in the Cartesian coordinate system with a radius of 4 units centered at the origin \((0, 0)\).
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Simplify this solution Bonus Knowledge
The polar equation \( r = 4 \) describes a circle with a radius of 4 centered at the origin in the polar coordinate system. When converting to rectangular coordinates, we use the relationship \( r^2 = x^2 + y^2 \). Substituting \( r = 4 \) gives \( 16 = x^2 + y^2 \), which is the standard form of the equation of a circle in rectangular coordinates. In the context of polar coordinates, the angle \( \theta \) doesn't affect the equation \( r = 4 \) because it represents a constant distance from the origin regardless of direction. This means every point on the circle is at exactly 4 units away from the origin, showcasing the beauty of symmetry in polar systems!
