Chapter 13, Problem 8RP

(a)

To determine

To find: The equation of the circle that passes through the given points.

Answer to Problem 8RP

The equation of the circle is ${\left(x-2\right)}^2+{\left(y+3\right)}^2=16$ .

Explanation of Solution

Given:

The given points are $\left(2,-3\right)$ and the radius of the circle is 4.

Calculation:

Consider the general form for the equation of the circle is,

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

Then, the equation of the circle that passes through the given points is,

  $\begin{array}{l}{\left(x-2\right)}^2+{\left(y-\left(-3\right)\right)}^2={\left(4\right)}^2\\ {\left(x-2\right)}^2+{\left(y+3\right)}^2=16\end{array}$

(b)

To determine

To find: The equation of the circle that passes through the given points.

Answer to Problem 8RP

The equation of the circle is $x^2+y^2=100$ .

Explanation of Solution

Given:

The given points are $\left(6,8\right)$ and the centre is at the origin.

Calculation:

Consider the general form for the equation of the circle with the centre at the origin is of the form,

  $x^2+y^2=r^2$

The radius of the circle is,

  $\begin{array}{l}{6}^2+8^2=r^2\\ r=10\end{array}$

Then, the equation of the circle that passes through the given points is,

  $x^2+y^2=100$

(c)

To determine

To find: The equation of the circle that passes through the given points.

Answer to Problem 8RP

The equation of the circle is ${\left(x-5\right)}^2+{\left(y\right)}^2=25$ .

Explanation of Solution

Given:

The end points of the diameter are $\left(0,0\right)$ and $\left(10,0\right)$ .

Calculation:

For the given end point the radius of the circle is,

  $\begin{array}{c}M=\left(\frac{10+0}{2},\frac{0+0}{2}\right)\\ =\left(5,0\right)\end{array}$

Consider the general form for the equation of the circle is,

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

Then, the equation of the circle that passes through the given points is,

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