Question

(a) Find the rectangular equation (in standard form) of the curve described by equation r2−2r(3sinθ−2cosθ)−3=0

(b) Write x2−y2=4 as a polar equation.

Answer 1

a) The rectangular equation of the curve is r^2 - 6rsinθ + 4rcosθ - 3 = 0. b) The polar equation of x^2 - y^2 = 4 is r^2cos(2θ) = 4.

Step-by-step explanation:

(a) The rectangular equation (in standard form) of the curve described by the equation r^2−2r(3sinθ−2cosθ)−3=0 can be found by expanding the equation and rearranging terms. First, distribute -2r to get -6rsinθ+4rcosθ. Then, move all terms to one side to get r^2 - 6rsinθ + 4rcosθ - 3 = 0. This is the rectangular equation in standard form.

(b) To write x^2−y^2=4 as a polar equation, we can use the polar coordinate transformations x = rcosθ and y = rsinθ. Substitute these values into the equation to get (rcosθ)^2 - (rsinθ)^2 = 4. Simplify the equation to get r^2(cos^2θ - sin^2θ) = 4. Since cos^2θ - sin^2θ = cos(2θ), the polar equation is r^2cos(2θ) = 4.