Answer: The correct option is (D) [tex]\dfrac{y}{x}=1.[/tex]
Step-by-step explanation: We are given to select the correct option that is true of the values of x and y in the diagram.
We can see from the figure that
the polar co-ordinates of the point (x, Y) are given by
[tex]r=1,~~~\theta=\dfrac{\pi}{4}.[/tex]
The relation between the Cartesian and polar co-ordinates are as follows :
[tex]x^2+y^2=r^2~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\\tan^{-1}\dfrac{y}{x}=\theta~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
Substituting the values of polar co-ordinates of the given point i the above equations, we get
[tex]x^2+y^2=1^2\\\\\Rightarrow x^2+y^2=1,[/tex]
and
[tex]\tan^{-1}\dfrac{y}{x}=\dfrac{\pi}{4}\\\\\\\Rightarrow \dfrac{y}{x}=\tan\dfrac{\pi}{4}\\\\\\\Rightarrow \dfrac{y}{x}=1.[/tex]
So, we get two equations :
[tex]x^2+y^2=1,~~~\dfrac{y}{x}=1.[/tex]
Since first equation is not in the given options, so the correct one is
(D) [tex]\dfrac{y}{x}=1.[/tex]
