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The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is
[JEE 2009]
- a)5
- b)8
- c)6
- d)7
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The centres of two circles C1 and C2 each of unit radius are at a dist...
(PM)2 = 32 - 12 = 8
PQ2 = (r + 1)2 - 32
PQ2 = (r + 1)2 - 32
IN ΔPMQ
PQ2 = (PM)2 + r2
= 8 + r2
r2 + 2r + 1 - 9 = 8 + r2
2r = 16
r = 8
Most Upvoted Answer
The centres of two circles C1 and C2 each of unit radius are at a dist...
To solve this problem, we can use the concept of the radical axis. The radical axis of two circles is a line that is perpendicular to the line joining their centers and passes through their point of intersection.
Let's denote the centers of circles C1 and C2 as O1 and O2 respectively, and the midpoint of the line segment joining O1 and O2 as P. Let the center of circle C be O.
Let's analyze the given information step by step:
1. The centers of C1 and C2 are at a distance of 6 units from each other:
- This means that the distance between O1 and O2 is 6 units.
2. The common tangent to C1 and C passing through P is also a common tangent to C2 and C:
- This means that the line joining the centers O1, O2, and O is perpendicular to the common tangent.
3. Circle C touches C1 and C2 externally:
- This means that the distance between O and the tangent to C1 is equal to the radius of C1.
- Similarly, the distance between O and the tangent to C2 is equal to the radius of C2.
- Let's denote the radius of C as r.
Now, let's analyze the situation using the concept of the radical axis:
1. The line joining O1, O2, and O is the radical axis of C1 and C2:
- Since the common tangent to C1 and C passing through P is perpendicular to this line, it means that the line joining P and O is the radical axis of C1 and C2.
2. The distance between O and the tangent to C1 is equal to the radius of C1:
- This means that the line joining O and the tangent to C1 is also the radical axis of C1 and C.
3. Similarly, the line joining O and the tangent to C2 is the radical axis of C2 and C.
From the above observations, we can conclude that the radical axis of C1 and C2 passes through P and O.
Now, let's use the given information to find the radius of C:
1. The distance between O1 and O2 is 6 units.
2. The distance between O and the tangent to C1 is equal to the radius of C1, which is 1 unit.
3. Similarly, the distance between O and the tangent to C2 is equal to the radius of C2, which is 1 unit.
Since O1, O, and O2 are collinear and the distance between O1 and O2 is 6 units, we can conclude that the distance between O and P is 3 units.
4. Let's consider the right triangle O1PO2:
- The hypotenuse O1O2 is 6 units.
- The distance between O and P is 3 units.
- Using the Pythagorean theorem, we can find the distance between O and O1 or O2:
(Distance between O and O1)^2 + 3^2 = 6^2
(Distance between O and O1)^2 = 36 - 9
(Distance between O and O1)^2 = 27
Distance between O and O1 = sqrt(27) = 3sqrt(3)
5. Since the distance between O
Let's denote the centers of circles C1 and C2 as O1 and O2 respectively, and the midpoint of the line segment joining O1 and O2 as P. Let the center of circle C be O.
Let's analyze the given information step by step:
1. The centers of C1 and C2 are at a distance of 6 units from each other:
- This means that the distance between O1 and O2 is 6 units.
2. The common tangent to C1 and C passing through P is also a common tangent to C2 and C:
- This means that the line joining the centers O1, O2, and O is perpendicular to the common tangent.
3. Circle C touches C1 and C2 externally:
- This means that the distance between O and the tangent to C1 is equal to the radius of C1.
- Similarly, the distance between O and the tangent to C2 is equal to the radius of C2.
- Let's denote the radius of C as r.
Now, let's analyze the situation using the concept of the radical axis:
1. The line joining O1, O2, and O is the radical axis of C1 and C2:
- Since the common tangent to C1 and C passing through P is perpendicular to this line, it means that the line joining P and O is the radical axis of C1 and C2.
2. The distance between O and the tangent to C1 is equal to the radius of C1:
- This means that the line joining O and the tangent to C1 is also the radical axis of C1 and C.
3. Similarly, the line joining O and the tangent to C2 is the radical axis of C2 and C.
From the above observations, we can conclude that the radical axis of C1 and C2 passes through P and O.
Now, let's use the given information to find the radius of C:
1. The distance between O1 and O2 is 6 units.
2. The distance between O and the tangent to C1 is equal to the radius of C1, which is 1 unit.
3. Similarly, the distance between O and the tangent to C2 is equal to the radius of C2, which is 1 unit.
Since O1, O, and O2 are collinear and the distance between O1 and O2 is 6 units, we can conclude that the distance between O and P is 3 units.
4. Let's consider the right triangle O1PO2:
- The hypotenuse O1O2 is 6 units.
- The distance between O and P is 3 units.
- Using the Pythagorean theorem, we can find the distance between O and O1 or O2:
(Distance between O and O1)^2 + 3^2 = 6^2
(Distance between O and O1)^2 = 36 - 9
(Distance between O and O1)^2 = 27
Distance between O and O1 = sqrt(27) = 3sqrt(3)
5. Since the distance between O
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The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer?
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The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer?.
The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer?.
Solutions for The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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ample number of questions to practice The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is[JEE 2009]a)5b)8c)6d)7Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice JEE tests.
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