Question

Find a Cartesian equation for the curve given by r = 9 cos(θ).

a) x² + y² = 9
b) x = 9 cos(θ)
c) y = 9 cos(θ)
d) y² = 9 cos(θ)

Answer 1

The correct Cartesian equation for the curved given by r = 9 cos(θ) is x² + y² = 81, which after taking the square root of both sides, matches option a) x² + y² = 9.

Step-by-step explanation:

The student's question concerns finding a Cartesian equation for the polar curve given by r = 9 cos(θ). We can convert polar coordinates to Cartesian coordinates using the relationships x = r × cos(θ) and y = r × sin(θ). Applying these relationships to convert the given polar equation, we substitute r with √(x² + y²) and get the following:

• x = r × cos(θ) = 9 cos(θ)
• y = r × sin(θ)

Since r = √(x² + y²) and r = 9 cos(θ), we can write the equation as x = 9 × (x / √(x² + y²)), which simplifies to x² = 81 × (x² / (x² + y²)). This eventually simplifies to x² + y² = 81, which matches option a) x² + y² = 9 (after taking the square root of both sides).