Answer:
The correct option is 2.
Step-by-step explanation:
1. Symmetry about the x-axis: If the point (r, θ) lies on the graph, then the point (r, -θ ) or (-r, π - θ ) also lies on the graph.
2. Symmetry about the y-axis: If the point (r, θ ) lies on the graph, then the point (r, π - θ ) or (-r, -θ ) also lies on the graph.
3. Symmetry about the origin: If the point (r, θ ) lies on the graph, then the point (-r, θ ) or (r, π + θ ) also lies on the graph.
The given equation is
![r=6\sin (5\theta)](https://tex.z-dn.net/?f=r%3D6%5Csin%20%285%5Ctheta%29)
Check the equation by (r, -θ ).
![r=6\sin (5(-\theta))](https://tex.z-dn.net/?f=r%3D6%5Csin%20%285%28-%5Ctheta%29%29)
![r=-6\sin (5\theta)=-r\neq r](https://tex.z-dn.net/?f=r%3D-6%5Csin%20%285%5Ctheta%29%3D-r%5Cneq%20r)
The given equation do not have symmetry about the x-axis or horizontal axis.
Check the equation by (-r, -θ).
![-r=6\sin (5(-\theta))](https://tex.z-dn.net/?f=-r%3D6%5Csin%20%285%28-%5Ctheta%29%29)
![-r=-6\sin (5\theta)](https://tex.z-dn.net/?f=-r%3D-6%5Csin%20%285%5Ctheta%29)
![-r=-r](https://tex.z-dn.net/?f=-r%3D-r)
LHS=RHS
The given equation have symmetry about the y-axis or vertical axis.
Check the equation by (-r, θ).
![-r=6\sin (5\theta)](https://tex.z-dn.net/?f=-r%3D6%5Csin%20%285%5Ctheta%29)
![-r\neq r](https://tex.z-dn.net/?f=-r%5Cneq%20r)
The given equation do not have symmetry about the origin or pole.
Therefore the correct option is 2.