Let's solve for [tex]\( r \)[/tex] in the given equation:
[tex]\[
\frac{f}{r^2} + a = n + k
\][/tex]
To isolate [tex]\( r \)[/tex], let's follow these steps:
1. Subtract [tex]\( a \)[/tex] from both sides:
[tex]\[
\frac{f}{r^2} = n + k - a
\][/tex]
2. Isolate [tex]\( r^2 \)[/tex] by multiplying both sides by [tex]\( r^2 \)[/tex]:
[tex]\[
f = (n + k - a)r^2
\][/tex]
3. Divide both sides by [tex]\( (n + k - a) \)[/tex]:
[tex]\[
r^2 = \frac{f}{n + k - a}
\][/tex]
4. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \pm \sqrt{\frac{f}{n + k - a}}
\][/tex]
The solutions for [tex]\( r \)[/tex] are:
[tex]\[
r = \pm \sqrt{\frac{f}{n + k - a}}
\][/tex]
Given the provided choices, the correct answer is:
[tex]\[
\boxed{\pm \sqrt{\frac{f}{n+k-a}}}
\][/tex]
