Question

Fill in the blanks to complete each of the following theorem statements:

For $a\ne 0$the graph of the equation$r=2a\sin \theta$is the...................... whose equation in rectangular coordinates is ....................

Short answer

The blanks are filled in order as:

1. Circle

2.$x^2+{\left(y-a\right)}^2=a^2$

Step-by-step solution

Step 1. Given information:

$r=2a\sin \theta$

Step 2. Convert the polar equation into rectangular form.

$$\begin{gathered} r=2a\sin \theta \\ r=2a\left(\frac{y}{r}\right) \\ {r}^2=2ay \\ {x}^2+y^2=2ay \\ {x}^2+y^2-2ay=0 \\ {x}^2+y^2-2ay+a^2=a^2 \\ {x}^2+{\left(y-a\right)}^2=a^2 \end{gathered}$$

This represents an equation of circle having center at$\left(0,a\right)$and radius$a$.