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Q. 39

Expert-verifiedFound in: Page 527

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 x ^{2} + y^{2 }= 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:

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

We have to find the radius of the circle.

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${r}_{2}=10$ .

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

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

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:

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

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

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

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.

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

We have to sketch the circles.

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:

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