Chapter 8, Problem 1RE

To determine

To find: The equation of the circle with center $\left(-2,3\right)$ and radius $5$. Draw the graph and also give the domain and range.

Answer to Problem 1RE

The equation of the circle is $\underset{\_}{{\left(x+2\right)}^2+{\left(y-3\right)}^2=25}$, the domain is $\underset{\_}{\left[-7,3\right]}$ and the range is $\underset{\_}{\left[-2,8\right]}$.

Explanation of Solution

Given:

The center of the circle is $\left(-2,3\right)$ and radius is $5$ unit.

Rules Used:

Centre-Radius form of the equation of a circle.

A circle with center $\left(h,k\right)$ and radius $r$ has equation ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$.

It follows that a circle with center at the origin $\left(0,0\right)$ and radius $r$ has equation $x^2+y^2=r^2$

Calculation:

The center-radius form of the equation of a circle is given below:

${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\mathrm{......}\left(1\right)$

The center $\left(h,k\right)$ of the circle is given as $\left(-2,3\right)$ and radius r as $5$ unit.

Therefore, the values of $h,k\text{and}r$ are $-2,3\text{and}5$.

Substitute $-2$ for $h$, $3$ for $k$ and $5$ for $r$ in the equation (1).

$\begin{array}{c}{\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\\ {\left[x-\left(-2\right)\right]}^2+{\left(y-3\right)}^2=5^2\\ {\left(x+2\right)}^2+{\left(y-3\right)}^2=25\end{array}$

As the center is $\left(-2,3\right)$ and radius is $5$ unit, the circle must pass through the four extreme points $\left(-2\pm 5,3\right)\text{and}\left(-2,3\pm 5\right)$ as shown in Table 1.

$x$	$y$
$3$	$3$
$-7$	$3$
$-2$	$8$
$-2$	$-2$

    Table 1

The graph for the equation ${\left(x+2\right)}^2+{\left(y-3\right)}^2=25$ is drawn in Figure 1.

The domain of the circle is set of all extreme values of $x$ and the range is the set of all extreme values of $y$.

Therefore, the domain is $\left[-7,3\right]$ and range is $\left[-2,8\right]$.

Thus, the equation of the circle is $\underset{\_}{{\left(x+2\right)}^2+{\left(y-3\right)}^2=25}$, the domain is $\underset{\_}{\left[-7,3\right]}$ and the range is $\underset{\_}{\left[-2,8\right]}$.