Question

Identify the conic represented by the polar equation$r=\frac{3}{1-2\cos \theta }$. Find the rectangular equation

Short answer

The given equation is of Hyperbola and the required equation in rectangular form is$\frac{y^2}{3}-\frac{{(x+2)}^2}{1}=1$

Step-by-step solution

Step 1. Given Information 

The given equation is$r=\frac{3}{1-2\cos \theta }$

Step 2. Calculation

As we know that the cartesian coordinate of a point is related with its polar coordinate as follows,

$x=r\cos \theta ,y=r\sin \theta \to r=\sqrt{x^2+y^2}$

Substitute the values in the given equation as follows,

$$\begin{gathered} r=\frac{3}{1-2\cos \theta } \\ r-2r\cos \theta =3 \\ \sqrt{x^2+y^2}-2x=3 \\ {\left(\sqrt{x^2+y^2}\right)}^2={\left(3+2x\right)}^2 \\ {x}^2+y^2=9+4x^2+12x \\ 3x^2-y^2+12x+9=0 \end{gathered}$$

From the cartesian form of the given polar equation, we can say that the given equation is of Hyperbola.

Step 3. Calculation

Convert the cartesian form of hyperbolic equation in the general form,

$$\begin{gathered} \frac{3x^2-y^2+12x+9}{3}=0 \\ {x}^2+4x-\frac{y^2}{3}=-3 \\ \left(x^2+4x+2^2\right)-\frac{y^2}{3}=-3+2^2 \\ {(x+2)}^2-\frac{y2}{3}=-1 \\ \frac{y^2}{3}-\frac{{(x+2)}^2}{1}=1 \end{gathered}$$