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A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)
- a)12
- b)14
- c)10
- d)8
Correct answer is option 'A'. Can you explain this answer?
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A circle of radius 4 cm is externally tangent to a circle of radius 9 ...
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A circle of radius 4 cm is externally tangent to a circle of radius 9 ...
Problem statement:
Find the length AB of a line that is tangent to two circles of radii 4 cm and 9 cm respectively and touches them at points A and B.
Solution:
Let O and O' be the centers of the circles of radii 4 cm and 9 cm respectively, and let C be the point of intersection of the line AB with the segment OO'.
[IMAGE HERE]
Since the line AB is tangent to both circles, we have
∠OAB = 90° and ∠O'BA = 90°.
Therefore, ∠AOB = ∠BO'C.
Also, since OA and OB are radii of the circle with center O, we have OA = OB = 4 cm.
Similarly, since OB and OC are radii of the circle with center O', we have OB = OC = 9 cm.
Let x be the length of AB. Then, we have
AC = AO + OC = 4 cm + 9 cm = 13 cm
and
BC = BO' - CO = 9 cm - 4 cm = 5 cm.
Applying the Pythagorean theorem to the right triangles AOC and BOC, we get
AC² = AO² + OC²
⇒ 169 = 16 + OC²
⇒ OC² = 153
⇒ OC = √153.
and
BC² = BO'² - CO²
⇒ 25 = 81 - CO²
⇒ CO² = 56
⇒ CO = √56.
Since ∠AOB = ∠BO'C, we have
∠AOC + ∠BO'C = 180°
⇒ ∠AOC = 180° - ∠BO'C
⇒ ∠AOC = 180° - ∠AOB.
Using the Law of Cosines in triangle AOC, we get
x² = AC² + AO² - 2(AC)(AO)cos∠AOC
⇒ x² = 169 + 16 - 2(13)(4)cos(180° - ∠AOB)
⇒ x² = 185 + 104cos∠AOB.
Using the Law of Cosines in triangle BOC, we get
x² = BC² + BO'² - 2(BC)(BO')cos∠BO'C
⇒ x² = 25 + 81 - 2(5)(9)cos∠AOB
⇒ x² = 106 - 90cos∠AOB.
Equating the two expressions for x², we get
185 + 104cos∠AOB = 106 - 90cos∠AOB
⇒ 194cos∠AOB = -79
⇒ cos∠AOB = -79/194.
Since ∠AOB is acute, we have
cos∠AOB = AB/OB
⇒ AB = OBcos∠AOB
⇒ AB = 9(-79/194)
⇒ AB = -36/97.
But AB is a length, so we take the absolute value to get
AB = 36/97.
Therefore, the length AB of the tangent
Find the length AB of a line that is tangent to two circles of radii 4 cm and 9 cm respectively and touches them at points A and B.
Solution:
Let O and O' be the centers of the circles of radii 4 cm and 9 cm respectively, and let C be the point of intersection of the line AB with the segment OO'.
[IMAGE HERE]
Since the line AB is tangent to both circles, we have
∠OAB = 90° and ∠O'BA = 90°.
Therefore, ∠AOB = ∠BO'C.
Also, since OA and OB are radii of the circle with center O, we have OA = OB = 4 cm.
Similarly, since OB and OC are radii of the circle with center O', we have OB = OC = 9 cm.
Let x be the length of AB. Then, we have
AC = AO + OC = 4 cm + 9 cm = 13 cm
and
BC = BO' - CO = 9 cm - 4 cm = 5 cm.
Applying the Pythagorean theorem to the right triangles AOC and BOC, we get
AC² = AO² + OC²
⇒ 169 = 16 + OC²
⇒ OC² = 153
⇒ OC = √153.
and
BC² = BO'² - CO²
⇒ 25 = 81 - CO²
⇒ CO² = 56
⇒ CO = √56.
Since ∠AOB = ∠BO'C, we have
∠AOC + ∠BO'C = 180°
⇒ ∠AOC = 180° - ∠BO'C
⇒ ∠AOC = 180° - ∠AOB.
Using the Law of Cosines in triangle AOC, we get
x² = AC² + AO² - 2(AC)(AO)cos∠AOC
⇒ x² = 169 + 16 - 2(13)(4)cos(180° - ∠AOB)
⇒ x² = 185 + 104cos∠AOB.
Using the Law of Cosines in triangle BOC, we get
x² = BC² + BO'² - 2(BC)(BO')cos∠BO'C
⇒ x² = 25 + 81 - 2(5)(9)cos∠AOB
⇒ x² = 106 - 90cos∠AOB.
Equating the two expressions for x², we get
185 + 104cos∠AOB = 106 - 90cos∠AOB
⇒ 194cos∠AOB = -79
⇒ cos∠AOB = -79/194.
Since ∠AOB is acute, we have
cos∠AOB = AB/OB
⇒ AB = OBcos∠AOB
⇒ AB = 9(-79/194)
⇒ AB = -36/97.
But AB is a length, so we take the absolute value to get
AB = 36/97.
Therefore, the length AB of the tangent
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A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)a)12b)14c)10d)8Correct answer is option 'A'. Can you explain this answer?
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A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)a)12b)14c)10d)8Correct answer is option 'A'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)a)12b)14c)10d)8Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)a)12b)14c)10d)8Correct answer is option 'A'. Can you explain this answer?.
A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)a)12b)14c)10d)8Correct answer is option 'A'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)a)12b)14c)10d)8Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle of radius 4 cm is externally tangent to a circle of radius 9 cm. A line is tangent to both circles and touches them at points A and B. Find the length AB? (in cm)a)12b)14c)10d)8Correct answer is option 'A'. Can you explain this answer?.
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