Question

Given the Iconic section r=2/1+2sinθ, identify the type of conic and find the verticies in both polar and cartesian coordinates.

Given the Iconic section ( r=frac{2}{1+2 sinθ} ), identify the type of conic and find the vertex (vertices) in both polar and cartesian coordinates. What kind of conic section is this?

Answer 1

The given equation represents a hyperbola. To find the vertices, convert the equation to Cartesian coordinates and solve for x.

Step-by-step explanation:

The given equation, r = √(2/(1 + 2sinθ)), represents a conic section known as a hyperbola. To identify the vertices, we need to convert the equation to rectangular (Cartesian) coordinates. Using the conversion formulas x = rcosθ and y = rsinθ, we substitute the given equation:

x = (√(2/(1 + 2sinθ)))cosθ

y = (√(2/(1 + 2sinθ)))sinθ

The vertices in Cartesian coordinates are the points where the hyperbola intersects the x-axis. Setting y=0, we can solve for x to find the x-coordinates of the vertices.

Answer: The given equation represents a hyperbola. The vertices in both polar and Cartesian coordinates can be determined by solving the equations mentioned above.