Two circles have the following equations: #(x +2 )^2+(y -1 )^2= 16 # and #(x +4 )^2+(y +7 )^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

circles overlap.

What we have to do here is compare the distance ( d ) between the centres of the circles with the sum or difference of the radii. There are 3 possible outcomes.

• If sum of radii > d , #color(blue)"then circles overlap"#
• If sum of radii < d , #color(blue)"then no overlap"#
• If difference of radii > d #color(red)"then 1 circle contains the other"#
The standard form of the #color(blue)"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 coordinates of the centre and r, the radius.
#(x+2)^2+(y-1)^2=16rArr" centre"=(-2,1),r=4#
#(x+4)^2+(y+7)^2=25rArr"centre"=(-4,-7),r=5#
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 coordinate points"#

The 2 points here are (-2 ,1) and (-4 ,-7) the centres of the circles.

#d=sqrt((-4+2)^2+(-7-1)^2)=sqrt(4+64)=sqrt68≈8.246#

sum of radii = 4 + 5 = 9

difference of radii = 5 - 4 = 1

Since sum of radii > d , then circles overlap graph{(y^2-2y+x^2+4x-11)(y^2+14y+x^2+8x+0)=0 [-40, 40, -20, 20]}

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Answer 2

To determine if one circle contains the other, you can compare the distance between their centers to the sum of their radii. If the distance between the centers is less than the sum of the radii, then one circle contains the other. Otherwise, they do not contain each other.

The center of the first circle is at (-2, 1) with a radius of 4, and the center of the second circle is at (-4, -7) with a radius of 5.

The distance between their centers can be found using the distance formula. If the distance between their centers is greater than the sum of their radii, then one circle does not contain the other.

If one circle doesn't contain the other, the greatest possible distance between a point on one circle and another point on the other can be found by subtracting the radii of the smaller circle from the distance between their centers.

Using the distance formula: [ \text{Distance} = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 } ]

After finding the distance, compare it with the sum of the radii. If the distance is greater than the sum of the radii, then the circles do not contain each other.

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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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