Find a polar equation for the curve represented by the given Cartesian equation $x^2+y^2=2$.

• A. $r=2$
• B. $r^2=2$
• C. $r=\sqrt{2}$
• D. $r^2=1$

Answer: (B)

Verified step by step guidance

1. Recall the relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
2. Substitute \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\) into the given Cartesian equation \(x^{2} + y^{2} = 2\) to express it in terms of \(r\) and \(\theta\).
3. After substitution, the equation becomes \((r \cos{\theta})^{2} + (r \sin{\theta})^{2} = 2\).
4. Use the Pythagorean identity \(\cos^{2}{\theta} + \sin^{2}{\theta} = 1\) to simplify the equation to \(r^{2} (\cos^{2}{\theta} + \sin^{2}{\theta}) = r^{2} = 2\).
5. Thus, the polar form of the curve is \(r^{2} = 2\). To express \(r\) explicitly, take the positive square root to get \(r = \sqrt{2}\).