The radius of the circles and distance between their centers. ( i ) ( x − 2 ) 2 + ( y − 1 ) 2 = 9 ( x − 6 ) 2 + ( y − 4 ) 2 = 16 ( i i ) x 2 + ( y − 2 ) 2 = 4 ( x − 5 ) 2 + ( y − 14 ) 2 = 9 ( i i i ) ( x − 3 ) 2 + ( y + 1 ) 2 = 1 ( x − 2 ) 2 + ( y − 2 ) 2 = 25 And interpret whether the circles intersect or not.

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Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.8, Problem 121E

  (a)

To determine

To calculate: The radius of the circles and distance between their centers.

  (i)(x2)2+(y1)2=9(x6)2+(y4)2=16

  (ii)x2+(y2)2=4(x5)2+(y14)2=9

  (iii)(x3)2+(y+1)2=1(x2)2+(y2)2=25

And interpret whether the circles intersect or not.

Expert Solution

Answer to Problem 121E

The radius of the circles and distance between their centers is provided below,

  Equation of circlesRadiusDistance betweentheir centersIntersect(i)(x2)2+(y1)2=9(x6)2+(y4)2=16345Yes(ii)x2+(y2)2=4(x5)2+(y14)2=92313No(iii)(x3)2+(y+1)2=1(x2)2+(y2)2=25153.2Yes

Explanation of Solution

Given information:

The pair of equation of circles,

  (i)(x2)2+(y1)2=9(x6)2+(y4)2=16

  (ii)x2+(y2)2=4(x5)2+(y14)2=9

  (iii)(x3)2+(y+1)2=1(x2)2+(y2)2=25

Formula used:

The standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Distance d(A,B) between two points A=(x1,y1) and B=(x2,y2) in the Cartesian plane is denoted by d(A,B)=(x2x1)2+(y2y1)2 .

Calculation:

Consider the equation,

  (i)(x2)2+(y1)2=9(x6)2+(y4)2=16 .

Rewrite the equation (x2)2+(y1)2=9 as,

  (x2)2+(y1)2=32

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x2)2+(y1)2=32 .

Here, h=2,k=1 and r=3 .

Therefore, center of circle is (2,1) and radius is 3 .

Next, rewrite the equation (x6)2+(y4)2=16 as,

  (x6)2+(y4)2=42

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x6)2+(y4)2=42 .

Here, h=6,k=4 and r=4 .

Therefore, center of circle is (6,4) and radius is 4 .

Now, distance between the centers of the circle is computed below,

Recall that the distance d(A,B) between two points A=(x1,y1) and B=(x2,y2) in the Cartesian plane is denoted by d(A,B)=(x2x1)2+(y2y1)2 .

Evaluate the distance between A(6,4) and B(2,1) .

  d(A,B)=(26)2+(14)2=16+9=25=5

Now, sum of radius of two circles is 3+4=7 .

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 (x2)2+(y1)2=9 and (x6)2+(y4)2=16 intersect each other.

Consider the equation,

  (ii)x2+(y2)2=4(x5)2+(y14)2=9 .

Rewrite the equation x2+(y2)2=4 as,

  (x0)2+(y2)2=22

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x0)2+(y2)2=22 .

Here, h=0,k=2 and r=2 .

Therefore, center of circle is (0,2) and radius is 2 .

Next, rewrite the equation (x5)2+(y14)2=9 as,

  (x5)2+(y14)2=32

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x5)2+(y14)2=32 .

Here, h=5,k=14 and r=3 .

Therefore, center of circle is (5,14) and radius is 3 .

Now, distance between the centers of the circle is computed below,

Recall that the distance d(A,B) between two points A=(x1,y1) and B=(x2,y2) in the Cartesian plane is denoted by d(A,B)=(x2x1)2+(y2y1)2 .

Evaluate the distance between A(0,2) and B(5,14) .

  d(A,B)=(50)2+(142)2=25+144=169=13

Now, sum of radius of two circles is 2+3=5 .

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 x2+(y2)2=4 and (x5)2+(y14)2=9 do not intersect each other.

Consider the equation,

  (iii)(x3)2+(y+1)2=1(x2)2+(y2)2=25 .

Rewrite the equation (x3)2+(y+1)2=1 as,

  (x3)2+(y(1))2=12

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x3)2+(y(1))2=12 .

Here, h=3,k=1 and r=1 .

Therefore, center of circle is (3,1) and radius is 1 .

Next, rewrite the equation (x2)2+(y2)2=25 as,

  (x2)2+(y2)2=52

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x2)2+(y2)2=52 .

Here, h=2,k=2 and r=2 .

Therefore, center of circle is (2,2) and radius is 5 .

Now, distance between the centers of the circle is computed below,

Recall that the distance d(A,B) between two points A=(x1,y1) and B=(x2,y2) in the Cartesian plane is denoted by d(A,B)=(x2x1)2+(y2y1)2 .

Evaluate the distance between A(3,1) and B(2,2) .

  d(A,B)=(23)2+(2+1)2=1+9=10=3.2

Now, sum of radius of two circles is 1+5=6 .

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 (x3)2+(y+1)2=1 and (x2)2+(y2)2=25 intersect each other.

Therefore, the above results are summarized as,

  Equation of circlesRadiusDistance betweentheir centersIntersect(i)(x2)2+(y1)2=9(x6)2+(y4)2=16345Yes(ii)x2+(y2)2=4(x5)2+(y14)2=92313No(iii)(x3)2+(y+1)2=1(x2)2+(y2)2=25153.2Yes

  (b)

To determine

To explain: Whether the circles intersect each other or not provided their radius and distance between their centers.

Expert Solution

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,

  (x2)2+(y1)2=9 and (x6)2+(y4)2=5 . Plot the circles on the Cartesian plane. Red curve represent the first circle and the blue curve represent the second circle.

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 1.8, Problem 121E

Rewrite the equation (x2)2+(y1)2=9 as,

  (x2)2+(y1)2=32

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x2)2+(y1)2=32 .

Here, h=2,k=1 and r=3 .

Therefore, center of circle is (2,1) and radius is 3 .

Next, rewrite the equation (x6)2+(y4)2=16 as,

  (x6)2+(y4)2=(5)2

Recall that the standard form of the equation of the circle is (xh)2+(yk)2=r2 , where (h,k) denote the center of the circle and r denote the radius.

Compare, (xh)2+(yk)2=r2 and (x6)2+(y4)2=42 .

Here, h=6,k=4 and r=5 .

Therefore, center of circle is (6,4) and radius is 5 .

Now, distance between the centers of the circle is computed below,

Recall that the distance d(A,B) between two points A=(x1,y1) and B=(x2,y2) in the Cartesian plane is denoted by d(A,B)=(x2x1)2+(y2y1)2 .

Evaluate the distance between A(6,4) and B(2,1) .

  d(A,B)=(26)2+(14)2=16+9=25=5

Now, sum of radius of two circles is 3+5=5.23 .

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 (x2)2+(y1)2=9 and (x6)2+(y4)2=5 intersect each other.

Now, in general terms if d is the distance between the center of two circles with radius r1 and r2 then two circles intersect each other at two points if distance between the two centers of the circle is less than sum of radius of two circles that is d<r1+r2 .

When distance between the two centers of the circle is equal to sum of radius of two circles that is d=r1+r2 , then two circles touch each other at one point.

When distance between the two centers of the circle is equal to sum of radius of two circles that is d>r1+r2 , then two circles do not intersect each other.

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