Question

the graph of $x^{2}-10x + y^{2}-10y=-1$ in the $xy$-plane is a circle. what is the length of the circles radius?

Explanation:

Step1: Complete the square for \(x\) terms

The standard form of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius.
For \(x^{2}-10x\), we have \((x - 5)^2=x^{2}-10x + 25\).
For \(y^{2}-10y\), we have \((y - 5)^2=y^{2}-10y+25\).
The given equation \(x^{2}-10x + y^{2}-10y=-1\) can be rewritten as \((x^{2}-10x + 25)+(y^{2}-10y + 25)=-1+25 + 25\).

Step2: Simplify the equation

Simplify the right - hand side: \(-1+25 + 25=49\).
The equation of the circle is \((x - 5)^2+(y - 5)^2=49\).
Since \(r^{2}=49\) (from the standard form \((x - a)^2+(y - b)^2=r^2\)), then \(r=\sqrt{49}\).

Answer:

\(7\)