To calculate: The radius of the circles and distance between their centers.
And interpret whether the circles intersect or not.
Answer to Problem 121E
The radius of the circles and distance between their centers is provided below,
Explanation of Solution
Given information:
The pair of equation of circles,
Formula used:
The standard form of the equation of the circle is
Distance
Calculation:
Consider the equation,
Rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Next, rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Now, distance between the centers of the circle is computed below,
Recall that the distance
Evaluate the distance between
Now, sum of radius of two circles is
When the distance between the two centers of the circle is less than sum of radius of two circles then the two circles intersect each other.
Therefore, the circles
Consider the equation,
Rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Next, rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Now, distance between the centers of the circle is computed below,
Recall that the distance
Evaluate the distance between
Now, sum of radius of two circles is
When the distance between the two centers of the circle is less than sum of radius of two circles then the two circles intersect each other.
Therefore, the circles
Consider the equation,
Rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Next, rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Now, distance between the centers of the circle is computed below,
Recall that the distance
Evaluate the distance between
Now, sum of radius of two circles is
When the distance between the two centers of the circle is less than sum of radius of two circles then the two circles intersect each other.
Therefore, the circles
Therefore, the above results are summarized as,
To explain: Whether the circles intersect each other or not provided their radius and distance between their centers.
Answer to Problem 121E
When the distance between the two centers of the circle is less than sum of radius of two circles then the two circles intersect each other.
Explanation of Solution
Given information:
The pair of equation of circles.
Consider the equation,
Rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Next, rewrite the equation
Recall that the standard form of the equation of the circle is
Compare,
Here,
Therefore, center of circle is
Now, distance between the centers of the circle is computed below,
Recall that the distance
Evaluate the distance between
Now, sum of radius of two circles is
When the distance between the two centers of the circle is less than sum of radius of two circles then the two circles intersect each other.
Therefore, the circles
Now, in general terms if d is the distance between the center of two circles with radius
When distance between the two centers of the circle is equal to sum of radius of two circles that is
When distance between the two centers of the circle is equal to sum of radius of two circles that is
Chapter 1 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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