Circle A has a radius of #2 # and a center of #(7 ,3 )#. Circle B has a radius of #3 # and a center of #(2 ,2 )#. If circle B is translated by ##, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Question

Circle A has a radius of #2 # and a center of #(7 ,3 )#.  Circle B has a radius of #3 # and a center of #(2 ,2 )#.  If circle B is translated by #<1 ,3 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Wyatt Anderson · Accepted Answer

The new centre of circle B will be #(3,5)#, which is #4.472# units from the centre of circle A. Since the sum of the radii of the two circles is #5# units, the circles overlap.

Translating circle B by #<1,3># simply requires adding 1 to the x-value and 3 to the y-value of the coordinates of its centre, so the new centre of circle B is #(3,5)#. The radii of the circles are 2 and 3 respectively, so if their centres are now less than 5 units apart they will overlap, but if they are more than 5 units apart they will not. To find the distance between the centres, #r#, we use an application of Pythagoras Theorem: #r = sqrt((y_2-y_1)^2+(x_2-x_1)^2) = sqrt((5-3)^2+(3-7)^2)# #= sqrt((2)^2+(-4)^2) = sqrt(4+16) = sqrt(20) = 4.472# units Because this is more than #5# units, the circles overlap. And hence, the question of the minimum distance between the circles does not arise. graph{((x-3)^2+(y-5)^2-9)((x-7)^2+(y-3)^2-4)((x-2)^2+(y-2)^2-0.02)((x-3)^2+(y-5)^2-0.02)=0 [-6.04, 13.96, -1.44, 8.56]}

Noah Atkinson · Answer

undefined To determine if circle B overlaps circle A after being translated by <1, 3>, we need to calculate the distance between the centers of the two circles and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, the circles overlap.

The distance between the centers of circle A and circle B after the translation is given by the distance formula:

$$
d = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}
$$

Where (x1, y1) are the coordinates of the center of circle A and (x2, y2) are the coordinates of the center of circle B after translation.

After translation, the center of circle B becomes (2 + 1, 2 + 3) = (3, 5).

Using the distance formula, we find:

$$
d = \sqrt{{(7 - 3)}^2 + {(3 - 5)}^2} = \sqrt{{4}^2 + {(-2)}^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47
$$

The sum of the radii of circle A and circle B is $2 + 3 = 5$.

Since the distance between the centers of the circles (4.47) is greater than the sum of their radii (5), the circles do not overlap.

To find the minimum distance between points on both circles, we subtract the sum of the radii from the distance between the centers:

$$
\text{Minimum distance} = d - (2 + 3) = \sqrt{20} - 5 \approx 4.47 - 5 = -0.53
$$

However, a negative distance is not meaningful in this context. So, the minimum distance between points on both circles is 0, indicating that the circles are tangent to each other.

Circle A has a radius of #2 # and a center of #(7 ,3 )#. Circle B has a radius of #3 # and a center of #(2 ,2 )#. If circle B is translated by #&lt;1 ,3 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #2 # and a center of #(7 ,3 )#. Circle B has a radius of #3 # and a center of #(2 ,2 )#. If circle B is translated by #<1 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?