Question

How do you find the eccentricity, directrix, focus and classify the conic section $r=\dfrac{0.8}{1-0.8\sin \theta }?$

Answer 1

We know that the given equation is in the polar form. We will convert the polar equation into the rectangular form. Then we will use the equation $r=\dfrac{ep}{1-e\sin \theta }$ to find the eccentricity, the directrix and the focus. Then using the eccentricity, we will find the conic section.

Complete step-by-step solution:
Let us consider the given polar equation $r=\dfrac{0.8}{1-0.8\sin \theta }.$
We know that there are some relations that connect the polar coordinates and the rectangular coordinates.
Here, we will use $x=r\cos \theta , y=r\sin \theta$ and $\sqrt{{{x}^{2}}+{{y}^{2}}}=r.$
Let us transpose the terms in the denominator from the right-hand side to the left-hand side.
We will get $r\left( 1-0.8\sin \theta \right)=0.8.$
This will give us $r-0.8r\sin \theta =0.8.$
Let us keep this aside.
We know that the given equation is similar to the equation $r=\dfrac{ep}{1-e\sin \theta }$ where $e$ is the eccentricity and $p$ is the distance of the directrix from the focus at pole. The negative sign implies that the directrix is below the focus and is parallel to the polar axis.
From this, when we compare the equations, we will get $e=0.8$ and $ep=0.8.$
Now, we have $e<1.$ Therefore the conic section is an ellipse.
Therefore, we will get $p=\dfrac{ep}{e}=\dfrac{0.8}{0.8}=1.$
That means the directrix is $1$ unit below the pole and is parallel to the polar axis.
Let us go back to the equation we have kept aside. Using the relations connecting the polar coordinates and the rectangular coordinates, we can convert the equation into $\sqrt{{{x}^{2}}+{{y}^{2}}}-0.8y=0.8.$
Hence the eccentricity is $e=0.8,$ the directrix is $y=-1$ and the focus is at pole $\left( 0,0 \right)$ and since the eccentricity $e$ is less than $1,$ the conic section is an ellipse.

Note: We can determine the conic section using the value of the eccentricity. If $e=0,$ then the conic section is a circle, if $e<1,$ then the conic section is an ellipse, if $e=1,$ then the conic section is a parabola, if $e>1,$ then the conic section is a hyperbola and if $e=\infty ,$ then the conic section is a line.