Question

Convert the following equation to Cartesian coordinates. Describe the resulting curve. r sin theta = 4 The Cartesian equation is [ ]. (Type an equation.) Describe the curve. Choose the correct answer below. A. The curve is a vertical line passing through (4,0). B. The curve is a line with slope 1/ 4 and y-intercept (0,0). C. The curve is a line with slope 1/4 and y-intercept (0,4). D. The curve is a horizontal line passing through (0,4).

Answer 1

Explanation:

To convert the equation r sin theta = 4 to Cartesian coordinates, we use the identities r^2 = x^2 + y^2 and y/x = tan theta. Substituting r sin theta = 4 into these identities, we get:

x^2 + y^2 = (r sin theta)^2 = 16

y/x = sin theta/ cos theta = tan theta

Squaring both sides of the second equation and substituting y^2/x^2 = 1 + tan^2 theta, we get:

y^2/x^2 = 1 + (y/x)^2

x^2 + y^2 = 16(1 + (y/x)^2)

Simplifying this equation, we get:

x^2 + y^2 = 16 + 4y^2/x^2

Multiplying both sides by x^2, we get:

x^2 y^2 + y^2 = 16x^2 + 4y^2

Bringing all the terms to one side, we get:

x^2 y^2 - 16x^2 = 3y^2

This is the Cartesian equation of the curve. To describe the curve, we can rewrite this equation as:

y^2/x^2 - 16/x^2 = 3

This is the equation of a hyperbola with center at the origin, vertical axis, and asymptotes given by y/x = ±4/sqrt(3).