Question

In the following exercises, ⓐ identify the center and radius, and ⓑ graph.

$x^2+y^2-4x+10y-7=0$

Short answer

(a) Center of the circle$x^2+y^2-4x+10y-7=0$ is$(2,-5)$and radius is$6$

(b) graph of the circle is

Step-by-step solution

Step 1. given data 

The given equation of the circle is

$x^2+y^2-4x+10y-7=0$

Step 2. Part (a) 

Group the x-terms and y-terms

$$\begin{gathered} x^2+y^2-4x+10y-7=0 \\ \left(x^2-4x\right)+\left(y^2+10y\right)=7 \end{gathered}$$

complete the squares

$$\begin{gathered} \left(x^2-4x+4\right)+\left(y^2+10y+25\right)=7+4+25 \\ {\left(x-2\right)}^2+{\left(y+5\right)}^2=36 \end{gathered}$$

Compare the equation with the standard equation of the circle

$$\begin{gathered} {\left(x-2\right)}^2+{\left(y-(-5)\right)}^2=6^2 \\ {\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2 \end{gathered}$$

so center$(h,k)=(2,-5)$

radius$r=6$

Step 3. Part (b) 

Plot the circle of center $(2,-5)$and radius$6$