Answer:
The correct answer option is [tex]r^2-6rsin(\theta)=8[/tex].
Step-by-step explanation:
We are given the following equation and we are to rewrite it in polar form:
[tex]x^2 + y^2 - 6y - 8 = 0[/tex]
[tex]\left \{ {{x^2+y^2=r^2} \atop {y=rsin \theta}} \right.[/tex]
Here, we need to make a perfect square trinomial:
[tex][r^2-6rsin(\theta)+[/tex] ___[tex]^2]=8[/tex]
[tex]6rsin(\theta)[/tex] ---> [tex]2 .r.3.sin \theta[/tex]
Completing the square to get:
[tex]r^2-6rsin(\theta)+[3sin(\theta)]^2-[3sin(\theta)]^2=8[/tex]
[tex][r-3sin(\theta)]^2=8+[3sin(\theta)]^2[/tex] --> [tex][r-3sin(\theta)]^2=8+9sin^2(\theta)[/tex]
[tex]r-3sin(\theta)=\sqrt{8+9sin^2(\theta)}[/tex] --> [tex]r=\sqrt{8+9sin^2(\theta)} +3sin(\theta)[/tex]
