Two circles have the following equations #(x -1 )^2+(y -7 )^2= 64 # and #(x +3 )^2+(y +3 )^2= 9 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Question

Two circles have the following equations #(x -1  )^2+(y -7   )^2= 64  # and #(x  +3   )^2+(y +3  )^2= 9 #.  Does one circle contain the other?  If not, what is the greatest possible distance between a point on one circle and another point on the other?

Autumn Armstrong · Accepted Answer

circles intersect

The standard form of the equation of a circle is : #color(red)(|bar(ul(color(white)(a/a)color(black)((x-a)^2+(y-b)^2=r^2)color(white)(a/a)|)))# where (a ,b) are the coords of the centre and r, the radius. #rArr(x-1)^2+(y-7)^2=64" has centre (1 ,7) and r=8"# and #(x+3)^2+(y+3)2=9" has centre (-3,-3)and r =3"# Now compare the distance (d) between the centres with the sum of the radii. • If sum of radii > d , then circles intersect • If sum of radii < d , then no intersection To calculate d , use the #color(blue)" distance formula"# #color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 coord points"# The 2 points here are the centres of the 2 circles. let #(x_1,y_1)=(1,7)" and " (x_2,y_2)=(-3,-3)# #d=sqrt((-3-1)^2+(-3-7)^2)=sqrt(16+100)≈10.77# sum of radii = 8 + 3 = 11 Since sum of radii > d , the circles intersect. graph{(y^2-14y+x^2-2x-14)(y^2+6y+x^2+6x+9)=0 [-40, 40, -20, 20]} Hence 1 circle is not contained within the other and the greatest distance between a point on 1 circle and a point on the other is d + sum of radii = 10.77 + 11 = 21.77

Caroline Aucoin · Answer

undefined The circles do not contain each other. The greatest possible distance between a point on one circle and another point on the other is the sum of their radii minus the distance between their centers.

Using the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$, which is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, we find the distance between the centers of the circles. Then, subtracting the sum of the radii from this distance gives us the greatest possible distance between points on the circles.

In this case, the distance between the centers of the circles is $\sqrt{(-3 - (-1))^2 + (-3 - 7)^2} = \sqrt{4^2 + (-10)^2} = \sqrt{16 + 100} = \sqrt{116}$.

The sum of the radii is $8 + 3 = 11$.

So, the greatest possible distance between points on the circles is $\sqrt{116} - 11$.