The equation of conic section can also be given by two dimensional vectors. The vector equation of conic must be a relation satisfied by position vectors of all the points on the conic. The position vector of a general point may be taken as $r$ . The eccentricity of the conic

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The given equation represents a conic section. By comparing it to the standard form of a conic in polar coordinates, $l / r = 1 + e cos θ$ , we can determine the eccentricity $e$ .

The provided image shows the equation:

$\frac{3}{r} = 2 + \frac{2 2}{3} cos θ$

To match the standard form, we need the right side to start with $1$ . Divide the entire equation by $2$ :

$\frac{3}{2 r} = 1 + \frac{2}{3} cos θ$

Rearranging to isolate $l / r$ :

$\frac{3/2}{r} = 1 + \frac{2}{3} cos θ$

Comparing this to $l / r = 1 + e cos θ$ , we identify the eccentricity $e$ .

Concepts used: Vector equation of a conic, standard polar form of a conic, eccentricity.

From the equation, $e = \frac{2}{3}$ .

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