# 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?

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

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

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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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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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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.

- A circle's center is at #(2 ,1 )# and it passes through #(3 ,7 )#. What is the length of an arc covering #(5pi ) /12 # radians on the circle?
- A triangle has corners at #(8 ,3 )#, #(2 ,4 )#, and #(7 ,2 )#. What is the area of the triangle's circumscribed circle?
- A triangle has corners at #(4 ,7 )#, #(8 ,9 )#, and #(3 ,5 )#. What is the area of the triangle's circumscribed circle?
- A circle has a chord that goes from #( 3 pi)/4 # to #(5 pi) / 4 # radians on the circle. If the area of the circle is #72 pi #, what is the length of the chord?
- In the figure AB is the diameter of the circle. PQ, RS are perpendicular to AB. Also #"PQ" = sqrt(18) "cm"#, #"RS" = sqrt(14) "cm"#. Find the diameter of this semicircle? Draw a semicircle of the same diameter and construct a square of area #20 "cm"^2#?

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