Two circles have the following equations #(x +2 )^2+(y -5 )^2= 16 # and #(x +4 )^2+(y -3 )^2= 49 #. 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 +2  )^2+(y -5   )^2= 16  # and #(x  +4   )^2+(y -3  )^2= 49 #.  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?

Avery Allen · Accepted Answer

Smaller circle (radius #4#) is contained in larger circle (radius #7#). The greatest possible distance between a point on one circle and another point on the other is #13.8284#.

The center of circle #(x+2)^2+(y-5)^2=16# is #(-2,5)# and radius is #4# and center of circle #(x+4)^2+(y-3)^2=49# is #(-4,3)# and radius is #7#. The distance between centers is #sqrt(((-4)-(-2))^2+(3-5)^2# = #sqrt(4+4)=sqrt8=2sqrt2=2.8284# If the radii of two circles is #r_1# and #r_2# and we also assume that #r_1>r_2# and the distance between centers is #d#, then A - if #r_1+r_2=d#, they touch each other externally and greatest possible distance between a point on one circle and another point on the other is #2(r_1+r_2)# and smallest possible distance between a point on one circle and another point on the other is #0#. B - if #r_1+r_2 < d#, they do not touch each other and are outside each other (i.e. one is not contained in other) and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest possible distance between a point on one circle and another point on the other is #d-r_1-r_2#. C - if #r_1+r_2 > d# and #r_1-r_2=d#, they touch each other internally and smaller circle is contained in other and greatest possible distance between a point on one circle and another point on the other is #2r_2# and smallest distance is #0#. D - if #r_1+r_2 > d# and #r_1-r_2>d#, smaller circle lies inside the larger circle and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest distance is #r_1-r_2-d#. E - if #r_1+r_2 > d# and #r_1-r_2 < d#, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d# and smallest distance is #0#. Now, here as #r_1+r_2=11>d=2.8284# and #r_1-r_2=3>d=2.8284#. Hence, smaller circle lies inside the larger circle and hence is contained in bigger circle. The greatest possible distance between a point on one circle and another point on the other is #r_1+r_2+d=7+4+2.8284=13.8284#. graph{(x^2+y^2+4x-10y+13)(x^2+y^2+8x-6y-24)=0 [-22, 18, -5.36, 14.64]}

Carson Arnold · Answer

undefined The two 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. In this case, the sum of the radii is $4 + 7 = 11$.

Avery Allen · Answer

undefined To determine if one circle contains the other, we can compare the radii of the two circles. The radius of the first circle is √16 = 4, and the radius of the second circle is √49 = 7.

Since the radius of the second circle (7) is greater than the radius of the first circle (4), the second circle cannot be contained within the first circle.

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

The center of the first circle is (-2, 5), and the center of the second circle is (-4, 3). Using the distance formula, we find the distance between their centers:

√((x2 - x1)^2 + (y2 - y1)^2) = √((-4 - (-2))^2 + (3 - 5)^2) = √(4^2 + (-2)^2) = √(16 + 4) = √20 = 2√5

The sum of the radii is 4 + 7 = 11.

Therefore, the greatest possible distance between a point on one circle and another point on the other circle is 2√5 - 11.