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Circle A has a center at #(-2 ,-7 )# and a radius of #2 #. Circle B has a center at #(-3 ,2 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

James Atkins

#color(blue)("Thus smallest distance between them is exactly "sqrt(82)-7)#

Let point 1 #->P_1=(x_1,y_1)=(-2,-7)#

Let point 2 #->P_2=(x_2,y_2)=(-3,2)#

Let distance between centres be #c#

#color(blue)("Sum of radii = 2 + 5 = 7")#

Distance between centres #->c= P_2-P_1#

So by Pythagoras:

#=>c=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#c=sqrt( [ color(white)(.)(-3)-(-2)color(white)(.)]^2+[color(white)(.)2-(-7)color(white)(.)]^2)#

#c=sqrt(1+81) = 9.055# to 3 decimal places

#color(blue)("Distance centre to centre = 9.055 to 3 decimal places")#

#color(brown)("Smallest distance between is (centre to centre) - (sum of radii)")#

#color(blue)("Thus smallest distance between them is exactly "sqrt(82)-7)#

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Answer 2

Aurora Ayala

The circles do not overlap. The smallest distance between them is 2 units.

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Answer from HIX Tutor

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.