Question

Find the eccentricity, directrix, focus and classify the conic section $r = \dfrac{8}{{4 - 1.6\sin \theta }}$?

Answer 1

Hint: The given equation deals with the concept of conic sections. Before starting to solve the given solution, we should first know what the conic section and its related terms. Here in this question, we are supposed to find the eccentricity, directrix, focus and classify the type of conic sections.

Complete step by step solution:
There are four types of conic equations: a parabola, an ellipse, a hyperbola and rectangular hyperbola.
Now, according to the question
We are given that $r = \dfrac{8}{{4 - 1.6\sin \theta }}$
The given conic section equation is similar to the standard equation $r = \dfrac{{ep}}{{1 - e\sin \theta }}$ where $e$ is the eccentricity of the conic whereas p is the distance of directrix from the focus at pole. The (-) sign means that the directrix is below the focus and is parallel to the polar axis.
$
   \Rightarrow r = \dfrac{8}{{4 - 1.6\sin \theta }} \\
   \Rightarrow r = \dfrac{{\dfrac{8}{4}}}{{\dfrac{{4 - 1.6\sin \theta }}{4}}} \\
   \Rightarrow r = \dfrac{2}{{1 - 0.4\sin \theta }} \\
 $
$\therefore e = 0.4$ and $ep = 2$ or $p = \dfrac{2}{{0.4}}$, $p = 5;e < 1$
Therefore, we can say that the conic is ellipse and the directrix is $5$ units below the pole and is parallel to the polar axis.
Now, we do rectangular conversion:
$
   \Rightarrow 4r - 1.6r\sin \theta = 8 \\
   \Rightarrow 4\sqrt {{x^2} + {y^2}} - 1.6y = 8 \\
 $
Therefore, we can conclude that
Eccentricity is $e = 0.4$
Directrix is $y = - 5$
Focus is at pole $\left( {0,0} \right)$
Conic is ellipse

Note: A conic section is defined as the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. The fixed point is called the focus. The fixed straight line is called the directrix. The constant ratio is known as the eccentricity denoted by e. The line passing through the focus and perpendicular to the directrix is called the axis and the point of intersection of a conic with its axis is known as vertex. The general equation of a conic section whose focus is $\left( {p,q} \right)$ and directrix is $lx + my + n = 0$:
$\left( {{l^2} + {m^2}} \right)\left[ {{{\left( {x - p} \right)}^2} + {{\left( {y - q} \right)}^2}} \right] = {e^2}{\left( {lx + my + n} \right)^2} = a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$