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Q. 39

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Geometry
Found in: Page 527
Geometry

Geometry

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

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Short Answer

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 ofx2+y2=25 is r1=5and the radius ofx−92+y−122=100 isr2=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

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

a.Step-1 – Given

The given equations arex2+y2=25 and x−92+y−122=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:x−a2+y−b2=r2 .

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

Comparing the equations, we get:

For the first equation: Radius =r1=5

And for the second equation: Radius =r2=10

Hence, radius ofx2+y2=25 isr1=5 and the radius ofx−92+y−122=100 isr2=10 .

b.Step-1 – Given

The given equations are x2+y2=25andx−92+y−122=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: x−a2+y−b2=r2.

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

Comparing the equations, we get:

For the first equation: Center =c1=0,0

And for the second equation: Center =c2=9,12

Using the distance formula:

c1c2=9−02+12−02c1c2=92+122c1c2=81+144c1c2=225c1c2=15

c.Step-1 – Given

The given equations arex2+y2=25 andx−92+y−122=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 arex2+y2=25 andx−92+y−122=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: PressY= button.

Step 3: Enter the expression x2+y2=25 and x−92+y−122=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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