FP2 Polar equations

So I have to find the polar equation from the following Cartesian equation : (x-1)^2+(y-1)^2=2 and I did this easily using the substitution of the results x=rcosΘ and y=rsinΘ to obtain the correct solution of r=2(cosΘ+sinΘ). But I also had to state
-1/4 pi <Θ< 3/4 pi and this is my question. On these sort of questions how do I find this interval of validity??? I have no idea about this so I full explanation answer would be useful if possible rather than helping me as I fully do not understand how to get this.
Also how do I find the polar equation of (1/x) + (1/y) = (1/a) where a x and y are all greater than 0. I am getting a=(rcosΘsinΘ)/(sinΘ+cosΘ) but the answer is r=a(secΘ+cosecΘ)

firstly there are some conventions when you are looking at polars

some define the polar plane in the θ "direction" as 0<= θ <2π
some define the polar plane in the θ "direction" as -π< θ <=π, as a complex argument



"a curve defined in an interval ..."
you do this by inspection and experience.
the interval they give there is the minimum interval in θ which traces one revolution of the circle but you can have a larger θ interval(all you do is you trace the curve over and over) or indeed a different interval


for your other question your equation can be simplified to the book's answer.

So there is no particular way of identifying the interval other than through inspection?

Okay thank you. So if I take say the angle Θ=pi which is outside of the range in the answer then what happens? Ie. Why is this not valid for the equation I determined?

So I have to find the polar equation from the following Cartesian equation : (x-1)^2+(y-1)^2=2 and I did this easily using the substitution of the results x=rcosΘ and y=rsinΘ to obtain the correct solution of r=2(cosΘ+sinΘ). But I also had to state
-1/4 pi <Θ< 3/4 pi and this is my question. On these sort of questions how do I find this interval of validity??? I have no idea about this so I full explanation answer would be useful if possible rather than helping me as I fully do not understand how to get this.