Given the polar equation $r=5\cos \left(\theta \right)$, which of the following is its Cartesian equation?

• A. $x^2+y^2=5x$
• B. $y=5x$
• C. $x^2+y^2=25$
• D. $x^2+y^2=5y$

Answer: (A)

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 \(r = 5 \cos(\theta)\), multiply both sides by \(r\) to get \(r^2 = 5r \cos(\theta)\).
3. Substitute \(r^2\) with \(x^2 + y^2\) and \(r \cos(\theta)\) with \(x\) to rewrite the equation as \(x^2 + y^2 = 5x\).
4. This equation is now in Cartesian form, representing a circle shifted along the x-axis.
5. You can further analyze or rearrange the equation to standard circle form if needed, but the key step is expressing the polar equation in terms of \(x\) and \(y\) using the coordinate conversions.