Answer
The answers:
(a) $r=2$ ${\ \ \ }$ (iii) Circle
(b) $\theta=2$ ${\ \ \ }$ (iv) Line through origin
(c) $r = 2\sec \theta $ ${\ \ \ }$ (i) Vertical line
(d) $r = 2\csc \theta $ ${\ \ \ }$ (ii) Horizontal line
Work Step by Step
(a) When $r=2$ , the distance of the curve is $2$ from the origin, so it is a circle of radius $2$. The answer is (iii) Circle.
(b) When $\theta=2$, it is a line through the origin that makes an angle $\theta=2$ with the $x$-axis. So, the answer is (iv) Line through origin.
(c) When $r = 2\sec \theta $, we have $2 = r\cos \theta = x$. The $x$-coordinate is $2$, a constant. So, it is a vertical line at $x=2$. The answer is (i) Vertical line.
(d) When $r = 2\csc \theta $, we have $2 = r\sin \theta = y$. The $y$-coordinate is $2$, a constant. So, it is a horizontal line at $y=2$. The answer is (ii) Horizontal line.
