Given the polar equation $r=7\cos \left(\theta \right)$, which of the following is the corresponding Cartesian equation?

• A. $x^2+y^2=7y$
• B. $y=7x$
• C. $x=7y$
• D. $x^2+y^2=7x$

Answer: (D)

Verified step by step guidance

1. Recall the relationships between polar and Cartesian coordinates: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), and also \(r^2 = x^2 + y^2\).
2. Given the polar equation \(r = 7 \cos(\theta)\), multiply both sides by \(r\) to get \(r^2 = 7r \cos(\theta)\).
3. Substitute \(r^2\) with \(x^2 + y^2\) and \(r \cos(\theta)\) with \(x\) to rewrite the equation in Cartesian form: \(x^2 + y^2 = 7x\).
4. This equation represents a circle in Cartesian coordinates, where the \(x\)-coordinate term appears on the right side.
5. Compare this derived Cartesian equation with the given options to identify the correct corresponding equation.