Final answer:
The polar equation r=4secθ is converted to Cartesian coordinates resulting in the line x = 4, representing a vertical line in the Cartesian plane.
Step-by-step explanation:
The given polar equation is r=4secθ. To convert this to Cartesian coordinates, we can use the relationships x = r × cosθ and y = r × sinθ. In terms of the Cartesian coordinate system, secθ is equivalent to the expression 1/cosθ,
so our equation becomes r = 4/cosθ. Substituting for r in terms of x and y, we have x = 4 since cosθ = x/r, and so 4/cosθ = 4r/x = r. Therefore, the graph of the equation is a vertical line where x = 4,
which is because secθ goes to infinity as θ approaches ± 90 degrees, indicating a vertical asymptote and a discontinuity in the graph at these points.