Question

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

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

Short answer

The equation of the given graph is$a)x^2+y^2=64$

Step-by-step solution

Step 1. Given Information

Given graph is

Given equations to match are:

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

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 ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$where $\left(h,k\right)$is the centre and $r$is the radius of the circle.

Here, from the graph the ends of the diameter of the circle are $\left(-8,0\right),\left(8,0\right)$.

So the centre is midpoint of the ends i.e. $\left(\frac{-8+8}{2},\frac{0+0}{2}\right)=\left(0,0\right)$

The radius of the circle is half the length of the diameter i.e. $\frac{8-\left(-8\right)}{2}=\frac{16}{2}=8$

Plugging the values in the equation of circle:

$$\begin{gathered} {\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2 \\ {\left(x-0\right)}^2+{\left(y-0\right)}^2=8^2 \\ {x}^2+y^2=64 \end{gathered}$$

Step 3. Conclusion

The equation of the given graph is$a)x^2+y^2=64$