Circle A has a radius of #2 # and a center of #(7 ,2 )#. Circle B has a radius of #3 # and a center of #(5 ,7 )#. If circle B is translated by #<-1 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
no overlap , ≈ 2.616 units.
• If sum of radii > d , then circles overlap
• If sum of radii < d , then no overlap
Before calculating d, we require to find the coordinates of the ' new' centre of circle B under the given translation which does not change the shape of the circle, only it's position.
The 2 points here are (7 ,2) and (4 ,9)
Sum of radii = radius of A + radius of B = 2 + 3 = 5
Since sum of radii < d, then no overlap
and min. distance between points = d - sum of radii
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To determine if circle B overlaps circle A after translation, calculate the distance between the centers of the circles. If this distance is less than the sum of the radii of both circles, they overlap. If not, the minimum distance between points on both circles is the difference between the distance between their centers and the sum of their radii.
Let ( C_A = (7, 2) ) be the center of circle A, and ( C_B = (5, 7) ) be the center of circle B before translation.
The distance between the centers of the circles is: [ d = \sqrt{(5 - 7)^2 + (7 - 2)^2} = \sqrt{4 + 25} = \sqrt{29} ]
The sum of the radii of both circles is: [ r_A + r_B = 2 + 3 = 5 ]
The translated center of circle B is ( C_B' = (5 - 1, 7 + 2) = (4, 9) ).
The distance between the centers of the translated circle B and circle A is: [ d' = \sqrt{(7 - 4)^2 + (2 - 9)^2} = \sqrt{9 + 49} = \sqrt{58} ]
Since ( \sqrt{58} > 5 ), the circles do not overlap after translation.
The minimum distance between points on both circles is the absolute difference between the distance between their centers and the sum of their radii: [ \text{Minimum distance} = \sqrt{58} - 5 ]
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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