Question

In the following exercises, match each graph to one of the following equations:

$$\begin{gathered} a)x^2+y^2=64 \\ b)x^2+y^2=49 \\ c){(x+5)}^2+{(y+2)}^2=4 \\ d){(x-2)}^2+{(y-3)}^2=9 \\ e)y=-x^2+8x-15 \\ f)y=6x^2+2x-1 \end{gathered}$$

Short answer

The equation of the given graph is$c){\left(x+5\right)}^2+{\left(y+2\right)}^2=4$

Step-by-step solution

Step 1. Given Information

Given graph is

Given equations to match are:

$$\begin{gathered} a)x^2+y^2=64 \\ b)x^2+y^2=49 \\ c){(x+5)}^2+{(y+2)}^2=4 \\ d){(x-2)}^2+{(y-3)}^2=9 \\ e)y=-x^2+8x-15 \\ f)y=6x^2+2x-1 \end{gathered}$$

The equation of the given graph among the given equations is to be determined.

Step 2. Equation of the graph

Given graph is a circle.

The standard form of a circle is ${(x-h)}^2+{(y-k)}^2=r^2$where $(h,k)$is the centre and $r$is the radius of the circle.

Here, from the graph the ends of the diameter of the circle are$(-3,-2),(-7,-2)$

So the centre is midpoint of the ends i.e.

$\left(\frac{-3+(-7)}{2},\frac{-2-2}{2}\right)=\left(-\frac{10}{2},\frac{-4}{2}\right)=\left(-5,-2\right)$

The radius of the circle is half the length of the diameter i.e

role="math" localid="1645617393863" $\left|\frac{-7-(-3)}{2}\right|=\left|\frac{-7+3}{2}\right|=\left|\frac{-4}{2}\right|=2$

Plugging the values in the equation of circle:

$$\begin{gathered} {(x-h)}^2+{(y-k)}^2=r^2 \\ {(x-(-5\left)\right)}^2+{(y-(-2\left)\right)}^2=2^2 \\ {(x+5)}^2+{(y+2)}^2=4 \end{gathered}$$

Step 3. Conclusion

The equation of the given graph is$c){(x+5)}^2+{(y+2)}^2=4$