Question

Eliminate $\theta$ if $x = r \cos \theta$ and $y = r \sin \theta$.

Hint:

To eliminate $\theta$ in polar equations, square both $x = r \cos \theta$ and $y = r \sin \theta$ and add them to use $\sin^2 \theta + \cos^2 \theta = 1$.

Answer 1

Step 1: Given equations.
We have \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \] Step 2: Square both equations and add them.
\[ x^2 = r^2 \cos^2 \theta \quad \text{and} \quad y^2 = r^2 \sin^2 \theta \] \[ x^2 + y^2 = r^2 (\cos^2 \theta + \sin^2 \theta) \] Step 3: Simplify using the trigonometric identity.
Since $\cos^2 \theta + \sin^2 \theta = 1$, we get \[ x^2 + y^2 = r^2 \] Step 4: Conclusion.
Thus, the required relation after eliminating $\theta$ is \[ \boxed{x^2 + y^2 = r^2} \] Correct Answer: $x^2 + y^2 = r^2$