# Thread: Largest distance between circles

1. The problem is this: what is the largest possible distance between a point on the circle (x - 3)^2 + y^2 = 4 and (x - 1)^2 + (y + 3)^2 = 25?
When I saw the problem, I immediately assumed it would be the distance between the circles plus the sum of the radii. This gives sqrt(13) + 7. However, the answer key said it was 12. Considering that sqrt(13) + 7 = 10.6055... which was relatively close to 12, I assumed there was a way of obtaining an even larger distance from two points that do not lie on the straight line connecting the radii. However, I could not figure out what these points were and to prove the distance was 12. Are there any ideas?
This question is from the PROBE I exam from 2011. Here is the solution guide, which does not show how they got their answers.
http://www.math.unl.edu/programs/mat...1probe1sol.pdf
Solution guides from other years do have the complete answer, such as the one from just last year, and I took this test in November.
http://www.math.unl.edu/programs/mat...ons_1111_4.pdf
These questions are for a mathematics competition for high school students in Nebraska (where I live).

2.

3. Looks wrong to me. Maybe that (x-1) was meant to be (x+1), then it'd work out, but (just drawing it out) it looks like the whole figure fits within a diameter 12 circle, so no two points within that circle should be that far apart.

4. If you draw a straight line passing through the radii of the circles, and define the line segment as the largest one there where it intersects the circles, you can make an even larger circle by using that segment as the diameter and it completely encloses the other two circles. You're right, MagiMaster. I think that (x - 1) should've been an (x + 1). Let's just hope that mistake wasn't on the actual test, or else nearly every student who took the exam got it wrong. The largest possible distance is indeed the distance between the centers plus the sum of the radii, so long as the circles do not intersect and neither of them is inside the other.

5. Your answer is correct. For the answer key to be correct, either the x–3 should be x+3 or the x–1 should be x+1.

6. Actually, these circles do intersect, but I don't think that changes the answer.

7. Uhh sorry but those arent circles theyre anything but. Is that a valid awnser? or do i ignore that I can never tell will you lose marks that way?

8. Originally Posted by anticorncob28
If you draw a straight line passing through the radii of the circles, and define the line segment as the largest one there where it intersects the circles, you can make an even larger circle by using that segment as the diameter and it completely encloses the other two circles. You're right, MagiMaster. I think that (x - 1) should've been an (x + 1). Let's just hope that mistake wasn't on the actual test, or else nearly every student who took the exam got it wrong. The largest possible distance is indeed the distance between the centers plus the sum of the radii, so long as the circles do not intersect and neither of them is inside the other.
Actually, even if the circles intersect, that still holds true, and if the line segments are treated as vectors it holds true if the circles are one completely enclosed by another, and by that I mean you subtract the distance between the centers from the sum of the radii, as it is effectively "backtracking" across both radii. Quite simply, both the largest and shortest distance between 2 circles is defined by a line segment lying along the line that intersects the centers of the 2 circles, except the shortest distance between intersecting circles. The shortest distance is the distance between the centers, subtracting the radii of the circles for circles completely separate of each other, for intersecting circles, the answer is always 0, and for 2 circles, where one lies within the other, the answer is the larger radius subtracting the sum of the smaller radius and distance between centers.

9. Uhh sorry but those arent circles theyre anything but. Is that a valid awnser? or do i ignore that I can never tell will you lose marks that way?
Are you that guy who's been suspended multiple time for posting nonsense? I recognize your username. Here's my response either way:
1) The test is multiple choice. You get 0 points for wrong answers, 1 point for questions left blank, and 4 points for questions answered correctly. So the best option in that case would to leave it blank. In fact they told us that if you're not more than 50% sure of your answer, don't fill it in, and NEVER randomly guess.
2) (More importantly) Any equation of the form (x - a)^2 + (y - b)^2 = c, where a, b, and c are real numbers and c is positive, forms a circle. These equations are in fact circles.

10. Originally Posted by fiveworlds
Uhh sorry but those arent circles theyre anything but.
They are circles.

Anticorncob28, maybe you're SUPPOSED to get one point for this question, which you should leave blank.

11. Okay, thanks for the help. Now I have another question. On the same test, can anybody figure out how you were supposed to do question #16? It is what is the difference between the maximum and minimum value of the function 3sin(2x) + 2cos(2x). It's easy to figure out with calculus, but this is supposed to be a test for high school students, so you shouldn't use anything more advanced than high school trigonometry. I personally was stumped by it.

12. Originally Posted by anticorncob28
Are you that guy who's been suspended multiple time for posting nonsense? I recognize your username.
Yep that's him, doesn't seem to have stopped him though...

13. Originally Posted by anticorncob28
It is what is the difference between the maximum and minimum value of the function 3sin(2x) + 2cos(2x). It's easy to figure out with calculus, but this is supposed to be a test for high school students, so you shouldn't use anything more advanced than high school trigonometry. I personally was stumped by it.
Express in the form and work out (you don’t need to work out ). Then the max and min values are so the difference is .

14. Thank you for the tip, Olinguito. I am a high schooler who is not particularly good at trigonometric identities, so there really was no hope for me working that out on my own without using calculus.

15. You’re welcome. Just remember that all expressions of the form can be rewritten as a , a simple sine function.

16. (3,0), 2 center and radius of first circle
(1,-3), 5 center and radius of second circle

m[(3,0)-(1,-3)]=(x,y)-(1,-3)

m(2,3)=(x-1,y+3)

m = m

(x-1)/2 = (y+3)/3 equation of line adjoining centers

y+3=(3/2)(x-1)

y=(3/2)x - 9/2 iii.

(x-3)^2 + ((3/2)x-9/2)^2=4 substitute iii. in equation of first circle
(x-1)^2 + ((3/2)x-9/2+3)^2=25 substitute iii. in equation of second circle

(x-3)^2 + ((3/2)x-9/2)^2=4
(x-1)^2 + ((3/2)x-3/2)^2=25

x^2-6x+9+(9/4)x^2-(27/2)x+81/4=4
x^2-2x+1+(9/4)x^2-(9/2)x+9/4=25

(1+9/4)x^2-(6+27/2)x+(5+81/4)=0
(1+9/4)x^2-(2+9/2)x+(9/4-24)=0

(13/4)x^2-(38/2)x+(101/4)=0 i.
(13/4)x^2-(13/2)x-(87/4)=0 ii.

solve i. for x and substitute in iii. and solve for y. This will result in 2 points on the first circle, p1 and p2.
solve ii. for x and substitute in iii. and solve for y. This will result in 2 points on the second circle, q1 and q2.

Solve |p1-q1|, |p1-q2|, |p2-q1|, |p2-q2| and select the greatest distance.

I drew a graph and this method works, but more simply put,

maximum distance = 5 + {(3-1)^2+(0-3)^2}^.5 + 2 = 7+sqrt(13)

Originally Posted by anticorncob28
The problem is this: what is the largest possible distance between a point on the circle (x - 3)^2 + y^2 = 4 and (x - 1)^2 + (y + 3)^2 = 25?
When I saw the problem, I immediately assumed it would be the distance between the circles plus the sum of the radii. This gives sqrt(13) + 7. However, the answer key said it was 12. Considering that sqrt(13) + 7 = 10.6055... which was relatively close to 12, I assumed there was a way of obtaining an even larger distance from two points that do not lie on the straight line connecting the radii. However, I could not figure out what these points were and to prove the distance was 12. Are there any ideas?
This question is from the PROBE I exam from 2011. Here is the solution guide, which does not show how they got their answers.
http://www.math.unl.edu/programs/mat...1probe1sol.pdf
Solution guides from other years do have the complete answer, such as the one from just last year, and I took this test in November.
http://www.math.unl.edu/programs/mat...ons_1111_4.pdf
These questions are for a mathematics competition for high school students in Nebraska (where I live).

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