Question

Find the center and radius of each circle, and sketch its graph.

(a)$x^2+y^2=25$

(b)${(x-2)}^2+{(y+1)}^2=9$

(c)$x^2+6x+y^2-2y+6=0$

Short answer

1. The centre of the circle is (0,0) and radius is 5
2. The centre is $(2,-1)$and radius of circle is 3
3. The centre of the circle is $(-3,1)$and radius of circle is 2

Step-by-step solution

Part a. Step 1. Determining general equation of circle.

An equation of the circle with centre $(h,k)$and radius r is ${(x-h)}^2+{(y-k)}^2=r^2$. This is called the standard form for the equation of the circle.

Part a. Step 2. Determining the centre and radius.

So, the centre is $(0,0)$ and radius is$\sqrt{25}=5$

Part a. Step 3. Plotting the graph.

Part b. Step 1. Determining general equation of circle.

An equation of the circle with centre $(h,k)$ and radius r is ${(x-h)}^2+{(y-k)}^2=r^2$. This is called the standard form for the equation of the circle.

Part b. Step 2. Determining the centre and radius.

So, the centre is $(2,-1)$and radius is$\sqrt{9}=3$

Part b. Step 3. Plotting the graph.

Part c. Step 1. Determining general equation of circle.

An equation of the circle with centre $(h,k)$ and radius r is ${(x-h)}^2+{(y-k)}^2=r^2$. This is called the standard form for the equation of the circle.

Part c. Step 2. Determining the centre and radius.

$$\begin{gathered} x^2+6x+y^2-2y+6=0 \\ {x}^2+6x+9-9+y-2y+1-1+6=0 \\ {(x+3)}^2+{(y-1)}^2=4 \end{gathered}$$=

So, the centre of circle is $(-3,1)$and radius of circle is 2

Part c. Step 3. Plotting the graph.

$x^2+6x+y^2-2y+6=0$

${(x+3)}^2+{(y-1)}^2=4$