Chapter 13.6, Problem 4PSA

(a)

To determine

To find: The equation of the circle from the given figure.

Answer to Problem 4PSA

The equation of the circle is $x^2+y^2=25$ .

Explanation of Solution

Given:

The given figure is shown in Figure 1

  

Figure 1

Calculation:

Consider the formula for the equation of the circle is of the form as,

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

Here, $\left(h,k\right)$ is the centre and $r$ is the radius.

Consider the formula for the distance between the two points is,

  $r=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

From the figure the centre of the circle is at $\left(0,0\right)$ and the radius of the circle by the distance formula is obtained as,

  $\begin{array}{c}r=\sqrt{{\left(-5-0\right)}^2+{\left(0-0\right)}^2}\\ =5\end{array}$

Then, substitute the values to obtain the equation for the circle as,

  $\begin{array}{l}{\left(x-0\right)}^2+{\left(y-0\right)}^2=5^2\\ x^2+y^2=25\end{array}$

(b)

To determine

To find: The equation of the circle from the given figure.

Answer to Problem 4PSA

The equation of the circle is ${\left(x+4\right)}^2+y^2=16$ .

Explanation of Solution

Given:

The given figure is shown in Figure 2

  

Figure 2

Calculation:

Consider the formula for the equation of the circle is of the form as,

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

Here, $\left(h,k\right)$ is the centre and $r$ is the radius.

Consider the formula for the distance between the two points is,

  $r=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

From the figure the centre of the circle is at $\left(-4,0\right)$ and the radius of the circle by the distance formula is obtained as,

  $\begin{array}{c}r=\sqrt{{\left(-4-0\right)}^2+{\left(0-0\right)}^2}\\ =4\end{array}$

Then, substitute the values to obtain the equation for the circle as,

  $\begin{array}{l}{\left(x-\left(-4\right)\right)}^2+{\left(y-0\right)}^2=4^2\\ {\left(x+4\right)}^2+y^2=16\end{array}$

(c)

To determine

To find: The equation of the circle from the given figure.

Answer to Problem 4PSA

The equation of the circle is ${\left(x-5\right)}^2+{\left(y-6\right)}^2=36$ .

Explanation of Solution

Given:

The given figure is shown in Figure 3

  

Figure 3

Calculation:

Consider the formula for the equation of the circle is of the form as,

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

Here, $\left(h,k\right)$ is the centre and $r$ is the radius.

Consider the formula for the distance between the two points is,

  $r=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

From the figure the centre of the circle is at $\left(5,6\right)$ and the radius of the circle by the distance formula is obtained as,

  $\begin{array}{c}r=\sqrt{{\left(5-5\right)}^2+{\left(6-0\right)}^2}\\ =6\end{array}$

Then, substitute the values to obtain the equation for the circle as,

  ${\left(x-5\right)}^2+{\left(y-6\right)}^2=36$