Question

10. Consider the circles with the following equations:

x^2+y^2=25 and
(x−9)^2+(y−12)^2=100.
a. What are the radii of the circles?
b. What is the distance between the centers of the circles?
c. Make a rough sketch of the two circles to explain why the circles must be tangent to one another.

Answer 1

a) 5 and 10

b) 15

c) The center-to-center distance is the sum of the radii, so the circles must be tangent.

Explanation:

a) Each equation is in standard form:

(x -h)^2 + (y -k)^2 = r^2

so the first circle has radius √25 = 5, and the second circle has radius √100 = 10. The radii of the circles are 5 and 10.

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b) The center-to-center distance can be found from the distance formula:

d = √((x2-x1)^2 +(y2-y1)^2) = √((9-0)^2 +(12-0)^2) = √(81 +144)

d = √225 = 15

The distance between centers if 15 units.

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c) The circles must be tangent because their center-to-center distance is the same as the sum of their radii.

5 + 10 = 15