Given the polar equation $r=9\cos \left(\theta \right)$, which of the following is its equivalent Cartesian equation?

• A. $x^2+y^2=9y$
• B. $y=9x$
• C. $x^2+y^2=9x$
• D. $x^2+y^2=81$

Answer: (C)

Verified step by step guidance

1. Recall the relationships between polar and Cartesian coordinates: \(r = \sqrt{x^{2} + y^{2}}\), \(x = r \cos(\theta)\), and \(y = r \sin(\theta)\).
2. Start with the given polar equation: \(r = 9 \cos(\theta)\).
3. Substitute \(\cos(\theta)\) with its Cartesian equivalent \(\frac{x}{r}\), so the equation becomes \(r = 9 \cdot \frac{x}{r}\).
4. Multiply both sides of the equation by \(r\) to eliminate the denominator: \(r^{2} = 9x\).
5. Replace \(r^{2}\) with \(x^{2} + y^{2}\) to get the Cartesian form: \(x^{2} + y^{2} = 9x\).