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Q. 39

Expert-verified
Found in: Page 527

### Geometry

Book edition Student Edition
Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen
Pages 227 pages
ISBN 9780395977279

# Find the radii of the circles x2 + y2 = 25 and (x - 9)2 + (y – 12)2 = 100.b. Find the distance between the centers of the circles.c. Explain why the circles must be externally tangent.d. Sketch the graphs of the circles.

1. Radius of${x}^{2}+{y}^{2}=25$ is ${r}_{1}=5$and the radius of${\left(x-9\right)}^{2}+{\left(y-12\right)}^{2}=100$ is${r}_{2}=10$ .

2. The distance is 15.

3. The distance between the centers of the circle is equal to sum of the radius. So, the circles must be externally tangent.

4. The graph is:

See the step by step solution

## a.Step-1 – Given

The given equations are${x}^{2}+{y}^{2}=25$ and ${\left(x-9\right)}^{2}+{\left(y-12\right)}^{2}=100$.

## Step-2 – To determine

We have to find the radius of the circle.

## Step-3 – Calculation

We’ll compare the given equations with the standard form:${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={r}^{2}$ .

Here (a, b) is the center and r being the radius.

Comparing the equations, we get:

For the first equation: Radius =${r}_{1}=5$

And for the second equation: Radius =${r}_{2}=10$

Hence, radius of${x}^{2}+{y}^{2}=25$ is${r}_{1}=5$ and the radius of${\left(x-9\right)}^{2}+{\left(y-12\right)}^{2}=100$ is${r}_{2}=10$ .

## b.Step-1 – Given

The given equations are ${x}^{2}+{y}^{2}=25$and${\left(x-9\right)}^{2}+{\left(y-12\right)}^{2}=100$ .

## Step-2 – To determine

We have to find the distance between the centers of the circles.

## Step-3 – Calculation

We’ll compare the given equations with the standard form: ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={r}^{2}$.

Here (a, b) is the center and r being the radius.

Comparing the equations, we get:

For the first equation: Center =${c}_{1}=\left(0,0\right)$

And for the second equation: Center =${c}_{2}=\left(9,12\right)$

Using the distance formula:

$\begin{array}{l}{c}_{1}{c}_{2}=\sqrt{{\left(9-0\right)}^{2}+{\left(12-0\right)}^{2}}\\ {c}_{1}{c}_{2}=\sqrt{{\left(9\right)}^{2}+{\left(12\right)}^{2}}\\ {c}_{1}{c}_{2}=\sqrt{81+144}\\ {c}_{1}{c}_{2}=\sqrt{225}\\ {c}_{1}{c}_{2}=15\end{array}$

## c.Step-1 – Given

The given equations are${x}^{2}+{y}^{2}=25$ and${\left(x-9\right)}^{2}+{\left(y-12\right)}^{2}=100$ .

## Step-2 – To determine

We have to explain why the circles must be externally tangent.

## Step-3 – Calculation

From part b we see that the distance between the centers = 15.

From part a,

Radius of the first circle = 5.

Radius of the second circle = 10.

So, 15 = 5 + 10.

We see that the distance between the centers of the circle is equal to sum of the radius. So, the circles must be externally tangent.

## d.Step-1 – Given

The given equations are${x}^{2}+{y}^{2}=25$ and${\left(x-9\right)}^{2}+{\left(y-12\right)}^{2}=100$ .

## Step-2 – To determine

We have to sketch the circles.

## Step-3 – Calculation

We’ll sketch the graph using a graphing utility.

Step 1: Press WINDOW button in order to access the Window editor.

Step 2: Press$\overline{)\text{Y=}}$ button.

Step 3: Enter the expression ${x}^{2}+{y}^{2}=25\text{and}{\left(x-9\right)}^{2}+{\left(y-12\right)}^{2}=100$ .

Step 4: Press GRAPH button to graph the function. Then adjust the windows according to the graph.

The obtained graph is: