Two circles have the following equations: #(x -1 )^2+(y -2 )^2= 9 # and #(x +6 )^2+(y +2 )^2= 25 #. 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?

Answer 1

The two circles are outside each other i.e. one circle is not contained in the other and greatest distance between a point on one circle and another point on the other is #16.06#

The center of #(x-1)^2+(y-2)^2=9# is#(1,2)# and radius is #3#.
and center of #(x+6)^2+(y+2)^2=25# is#(-6,-2)# and radius is #5#.
The distance between centers is #sqrt((1-(-6))^2+(2-(-2))^2)#
= #sqrt(7^2+4^2)=sqrt(49+16)=sqrt65=8.06#
Hence sum of raadii is #8# and is less than distance between their centers.

Hence the two circles are outside each other i.e. one circle is not contained in the other.

and greatest distance between a point on one circle and another point on the other is#3+5+8.06=16.06#

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graph{((x-1)^2+(y-2)^2-9)((x+6)^2+(y+2)^2-25)=0 [-2.853, -0.353, -0.085, 1.165]}

graph{((x-1)^2+(y-2)^2-9)((x+6)^2+(y+2)^2-25)=0 [-12.29, 7.71, -5.34, 4.66]}

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Answer 2

To determine if one circle contains the other, you can compare the distances between their centers and the radii. If the distance between the centers of the circles is less than the sum of their radii, then one circle contains the other.

The centers of the given circles are at (1, 2) and (-6, -2), and their radii are 3 and 5 respectively.

The distance between the centers of the circles is ( \sqrt{(1 - (-6))^2 + (2 - (-2))^2} = \sqrt{7^2 + 4^2} = \sqrt{65} \approx 8.06 ).

The sum of the radii is ( 3 + 5 = 8 ).

Since the distance between the centers (( \sqrt{65} )) is greater than the sum of the radii (8), one circle does not contain the other.

To find the greatest possible distance between a point on one circle and another point on the other circle, you can find the distance between their centers and subtract the sum of their radii:

[ \text{Greatest possible distance} = \sqrt{65} - (3 + 5) = \sqrt{65} - 8 \approx 0.06 ]

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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.

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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