Question

R = 6 / 2sinθ Find the eccentricity and identify the conic.

Answer 1

The eccentricity is 1/2, confirming that the conic section is an ellipse.

How to find eccentricity and conic?

The given equation is in polar form for a conic section:

`\[ r = (a)/(1 - e \cdot \cos(\theta)) \]`

Comparing this with the given equation
`\( r = (6)/(2\sin(\theta)) \)`, identify the corresponding parameters:

a = 6

`\[ e = (1)/(2) \]`

The eccentricity (e) is given by the ratio of the distance from the focus to the directrix to the distance from the origin to the directrix. In this case,
`\(e = (1)/(2)\)`, which is less than 1, indicating an ellipse.

So, the conic section represented by the equation
`\( r = (6)/(2\sin(\theta)) \)` is an ellipse with an eccentricity of
`\((1)/(2)\)`.