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It is known that two circles of radii 3 cm and 6 cm have at most 1 common tangent. Let d and D be the minimum and maximum possible distances between their centres. What is the value, in cm, of (D - d)?
Correct answer is '3'. Can you explain this answer?
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It is known that two circles of radii 3 cm and 6 cm have at most 1 com...
If the two circles have at most 1 common tangent, they can either have no common tangents or one common tangent. They will have no common tangent if one circle is completely within the other with no point of contact, and one common tangent if one circle is inside the other, but there is one point of contact.
To find d (the minimum distance between centres) we observe that the least distance between centres will occur when the circles are concentric, so d = 0.
To find d (the minimum distance between centres) we observe that the least distance between centres will occur when the circles are concentric, so d = 0.
Clearly, the centres will be farthest apart when the circles are tangent to each other (because, in both our cases, we know that one circle is interior to the other).
Thus, D is equal to the difference in the radii o f the circles = 6 - 3 = 3 cm Further, (D - d) = 3 - 0 = 3 cm Answer: 3
Thus, D is equal to the difference in the radii o f the circles = 6 - 3 = 3 cm Further, (D - d) = 3 - 0 = 3 cm Answer: 3
Most Upvoted Answer
It is known that two circles of radii 3 cm and 6 cm have at most 1 com...
Given:
- Two circles of radii 3 cm and 6 cm
To find:
- The minimum and maximum possible distances between their centers
Solution:
Step 1: Understand the Problem
- We need to find the minimum and maximum possible distances between the centers of two circles.
- The circles have radii of 3 cm and 6 cm.
- The problem states that the circles have at most 1 common tangent.
Step 2: Visualize the Circles
- Let's visualize the two circles and their centers.
- The circle with a radius of 3 cm will be smaller and the circle with a radius of 6 cm will be larger.
Step 3: Tangents and Centers
- In order to find the minimum and maximum distances, we need to consider the position of the common tangent(s) between the circles.
- Since the circles have at most 1 common tangent, there are two possible cases to consider:
1. The circles do not intersect.
2. The circles intersect.
Step 4: Case 1 - Circles Do Not Intersect
- In this case, the distance between the centers of the circles is equal to the sum of their radii.
- The minimum distance (d) will occur when the circles are externally tangent.
- The distance between the centers will be 3 cm + 6 cm = 9 cm.
- The maximum distance (D) will occur when the circles are internally tangent.
- The distance between the centers will be 6 cm - 3 cm = 3 cm.
Step 5: Case 2 - Circles Intersect
- In this case, the distance between the centers will be less than the sum of their radii.
- The minimum distance (d) will occur when the circles are externally tangent.
- The distance between the centers will be 3 cm + 6 cm = 9 cm.
- The maximum distance (D) will occur when one circle is completely inside the other.
- The distance between the centers will be 6 cm - 3 cm = 3 cm.
Step 6: Comparison and Final Answer
- Comparing the minimum and maximum distances in both cases:
- Minimum distance (d) = 9 cm
- Maximum distance (D) = 3 cm
- The difference between the maximum and minimum distances is 9 cm - 3 cm = 6 cm.
- Therefore, the value of (D - d) is 6 cm, which contradicts the given correct answer of 3 cm.
- There might be an error in the given correct answer, or there could be a misunderstanding in the question or solution approach.
- Two circles of radii 3 cm and 6 cm
To find:
- The minimum and maximum possible distances between their centers
Solution:
Step 1: Understand the Problem
- We need to find the minimum and maximum possible distances between the centers of two circles.
- The circles have radii of 3 cm and 6 cm.
- The problem states that the circles have at most 1 common tangent.
Step 2: Visualize the Circles
- Let's visualize the two circles and their centers.
- The circle with a radius of 3 cm will be smaller and the circle with a radius of 6 cm will be larger.
Step 3: Tangents and Centers
- In order to find the minimum and maximum distances, we need to consider the position of the common tangent(s) between the circles.
- Since the circles have at most 1 common tangent, there are two possible cases to consider:
1. The circles do not intersect.
2. The circles intersect.
Step 4: Case 1 - Circles Do Not Intersect
- In this case, the distance between the centers of the circles is equal to the sum of their radii.
- The minimum distance (d) will occur when the circles are externally tangent.
- The distance between the centers will be 3 cm + 6 cm = 9 cm.
- The maximum distance (D) will occur when the circles are internally tangent.
- The distance between the centers will be 6 cm - 3 cm = 3 cm.
Step 5: Case 2 - Circles Intersect
- In this case, the distance between the centers will be less than the sum of their radii.
- The minimum distance (d) will occur when the circles are externally tangent.
- The distance between the centers will be 3 cm + 6 cm = 9 cm.
- The maximum distance (D) will occur when one circle is completely inside the other.
- The distance between the centers will be 6 cm - 3 cm = 3 cm.
Step 6: Comparison and Final Answer
- Comparing the minimum and maximum distances in both cases:
- Minimum distance (d) = 9 cm
- Maximum distance (D) = 3 cm
- The difference between the maximum and minimum distances is 9 cm - 3 cm = 6 cm.
- Therefore, the value of (D - d) is 6 cm, which contradicts the given correct answer of 3 cm.
- There might be an error in the given correct answer, or there could be a misunderstanding in the question or solution approach.
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Community Answer
It is known that two circles of radii 3 cm and 6 cm have at most 1 com...
One common tangent means smaller circle is inscribed in side bigger circle touching at one point.now, max and min will be same as 6-3=3 would be distance between there centre
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It is known that two circles of radii 3 cm and 6 cm have at most 1 common tangent. Let d and D be the minimum and maximum possible distances between their centres. What is the value, in cm, of (D - d)?Correct answer is '3'. Can you explain this answer?
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