Finding the Unknown Coordinate when given Distances from two points I have this problem that I don't know how to do.

Locate the two points that are $5$ units from $(5,-3)$ and square root of $41$ units from$(-2,6)$


You need the intersection of two circles.
Let's say $A=(5,-3)$ and $B=(-2,6)$, $r_1=5$ and $r_2=\sqrt{41}$.
Now
$$c_1:(x-5)^2+(y+3)^2=5^2 \quad and \quad c_2:(x+2)^2+(y-6)^2=\sqrt{41}^2$$
Now you have to find you radical axis of $c_1,c_2$:
\begin{eqnarray*}
c_1-c_2&:x^2-10x+y^2+6y+9=0 \\ 
&-[x^2+4x+y^2-12y-1=0]\\
&\Rightarrow z(x)=\frac 7 9 x - \frac 5 9
\end{eqnarray*}
With $z(x) \cap c_1$ you'll get
$$(x-5)^2+(z(x)+3)^2=25$$ with $L=\lbrace \frac{121}{65}, 2 \rbrace$ for $x$. So your two points are $S_1(\frac{121}{65};\frac{58}{65})$ and $S_2(2;1)$
