Question

. Prove that the graph of the equation $r=kcoesc\theta$, $0<\theta <\pi$, is a horizontal line for any value of $k\ne 0$

Short answer

Equation in rectangular form is $y=k$, hence it represent a horizontal line which is at a distance of$k$units from$x-$axis.

Step-by-step solution

Step 1. Given information

The equation in polar form :$r=k\cos ec\theta$

$0<\theta <\pi$

Step 2. Convert the polar form of equation into rectangular form:

Since in $y=r\sin \theta$

$$\begin{gathered} \sin \theta =\frac{y}{r} \\ \cos ec\theta =\frac{r}{y} \end{gathered}$$

$$\begin{gathered} r=k\cos ec\theta \\ r=k\left(\frac{r}{y}\right) \\ y=k \end{gathered}$$

This is an equation of horizontal line .