Question

Convert the following equation to Cartesian coordinates. Describe the resulting curve.

r= 8 cos θ + 2 sin θ
Write the Cartesian equation. Describe the curve.
A. The curve is a horizontal line with y-intercept at the point.
B. The curve is a cardioid with symmetry about the y-axis.
C. The curve is a vertical line with x-intercept at the point
E. The curve is a cardioid with symmetry about the x-axis.

Answer 1

To convert the equation r = 8cos(θ) + 2sin(θ) to Cartesian coordinates, we use the relationships x = rcos(θ) and y = rsin(θ). The resulting Cartesian equation is x = 8cos²(θ) - 2sin²(θ) and y = 8sin(θ)cos(θ) + 2sin²(θ). The curve is a cardioid with symmetry about the x-axis.

Step-by-step explanation:

To convert the equation r = 8cos(θ) + 2sin(θ) to Cartesian coordinates, we can use the relationships x = rcos(θ) and y = rsin(θ). Substituting these expressions into the given equation, we get x = 8cos(θ)cos(θ) + 2sin(θ)cos(θ) and y = 8cos(θ)sin(θ) + 2sin²(θ). Simplifying further, we have x = 8cos²(θ) - 2sin²(θ) and y = 8sin(θ)cos(θ) + 2sin²(θ) as the Cartesian equation. The resulting curve is a cardioid with symmetry about the x-axis.