Two circles have the following equations #(x -1 )^2+(y -2 )^2= 64 # and #(x +7 )^2+(y +2 )^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 -2   )^2= 64  # and #(x  +7   )^2+(y +2  )^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?

Noah Atkinson · Accepted Answer

circles overlap.

What we have to do here is #color(blue)"compare"# the distance ( d) between the centres of the circles to the #color(blue)"sum/difference of the radii"# • When radii add up to more than d, circles overlap. • There is no overlap if the sum of the radii is less than d. • One circle contained in another if the difference in radii is greater than d utilizing an equation to determine the circles' centers and radii. #color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)((x-a)^2+(y-b)^2=r^2)color(white)(2/2)|)))# where (a ,b) are the coordinates of the centre and r, the radius. #(x-1)^2+(y-2)^2=64" has centre (1,2) and r = 8"# #(x+7)^2+(y+2)^2=9" has centre (-7,-2) and r = 3"# To find d, use the #color(blue)"distance formula"# #color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))# where # (x_1,y_1),(x_2,y_2)" are 2 coordinate points"# Here, there are two points: (1,2) and (-7,-2). let # (x_1,y_1)=(1,2)" and " (x_2,y_2)=(-7,-2)# #d=sqrt((-7-1)^2+(-2-2)^2)=sqrt80≈8.944# Radius total = 8 + 3 = 11 Circles overlap in the graph {(y^2-4y+x^2-2x-59)(y^2+4y+x^2+14x+44)=0 [-40, 40, -20, 20]} because the sum of the radii is greater than d.

Violet Aldrich · Answer

undefined To determine if one circle contains the other, we can compare their radii. The radius of the first circle is $ \sqrt{64} = 8 $, and the radius of the second circle is $ \sqrt{9} = 3 $. Since the radius of the first circle (8) is greater than the radius of the second circle (3), the second circle cannot contain the first.

To find the greatest possible distance between a point on one circle and another point on the other, we need to find the distance between the centers of the circles and subtract the sum of their radii.

The center of the first circle is at (1, 2) and the center of the second circle is at (-7, -2). Using the distance formula, the distance between the centers is:

$$ \sqrt{(1 - (-7))^2 + (2 - (-2))^2} = \sqrt{64 + 16} = \sqrt{80} $$

Now, subtract the sum of their radii:

$$ \sqrt{80} - (8 + 3) = \sqrt{80} - 11 $$

Thus, the greatest possible distance between a point on one circle and another point on the other is $ \sqrt{80} - 11 $.