Chapter 11, Problem 17STP

(a)

To determine

Find the coordinate of the center of the circle. Explain your method.

Answer to Problem 17STP

The coordinate of the center is (2, − 4)

Explanation of Solution

Given:

The end points of a diameter of a circle are at (− 1, 0) and (5, − 8),

What are the coordinate of the center of the circle? Explain your method.

Concept Used:

Using mid − point theorem

Let $(x_1,y_1)\text{and}(x_2,y_2)$ are two points. The Mid − point ${\text{P}}_{(x,y)}$

  $x=\frac{1}{2}(x_1+x_2);y=\frac{1}{2}(y_1+y_2)$

Calculation:

Using mid − point theorem

Let $(x_1,y_1)\text{and}(x_2,y_2)$ are two points. The Mid − point ${\text{P}}_{(x,y)}$

  $x=\frac{1}{2}(x_1+x_2);y=\frac{1}{2}(y_1+y_2)$

Coordinates of the center:

  $\begin{array}{l}x=\frac{1}{2}(x_1+x_2);y=\frac{1}{2}(y_1+y_2)\\ x_1=-1,y_1=0\text{and}{x}_2=5,y_2=-8\\ \text{Center}(x,y)=\frac{1}{2}(x_1+x_2);\frac{1}{2}(y_1+y_2)=\frac{1}{2}(-1+5);\frac{1}{2}(0-8)\Rightarrow (2,-4)\\ \text{Center}(x,y)=(2,-4)\end{array}$

Thus, coordinate of the center is (2, − 4)

(b)

To determine

Find the radius of the circle. Explain your method.

Answer to Problem 17STP

The radius is 5.

Explanation of Solution

Given:

The end points of a diameter of a circle are at (− 1, 0) and (5, − 8)

Find the radius of the circle. Explain your method.

Concept Used:

Here we are using the distance formula. Distance between two points: $(x_1,y_1);(x_2,y_2)$

  $\text{Distance}=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$

Calculation:

  $\begin{array}{l}\text{Distance}=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}\\ x_1=2,y_1=-4\text{and}{x}_2=5,y_2=-8\\ \text{Radius}:\sqrt{{(2-5)}^2+{(-4-(-8))}^2}=\sqrt{9+{(4)}^2}=\sqrt{25}=5\end{array}$

Thus, the radius is 5.

(c)

To determine

Write an equation of the circle

Answer to Problem 17STP

The equation of circle is: ${(x-2)}^2+{(y+4)}^2=25$

Explanation of Solution

Given:

The end points of a diameter of a circle are at (− 1, 0) and (5, − 8),

Write an equation of the circle.

Concept Used:

Equation of circle: ${(x-h)}^2+{(y-k)}^2=r^2$ where center is ( h, k ) and r represents the radius.

Calculation:

Equation of circle: ${(x-h)}^2+{(y-k)}^2=r^2$

  $\begin{array}{l}\text{Center}:h=2,k=-4;\text{Radius}=r=5\\ $ {(x-h)}^2+{(y-k)}^2=r^2$$ {(x-2)}^2+{(y-(-4))}^2=5^2$$ {(x-2)}^2+{(y+4)}^2=25$\end{array}$

Thus, the equation of circle is: ${(x-2)}^2+{(y+4)}^2=25$