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Two circles of radii 5 cm and 3 cm touch externally, then the ratio in which the direct common tangent to the circles divides externally the line joining the centres of the circles is: (SSC CHSL 2015)
- a)2.5 : 1.5
- b)1.5 : 2.5
- c)3 : 5
- d)5 : 3
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Two circles of radii 5 cm and 3 cm touch externally, then the ratio in...
Let PB = x cm.
In ∠PQB and PRA,
∠Q = ∠R = 90°, ∠P = ∠P {common}
∴ ΔPQB ~ ΔPRA {AA criteria}
Now, point P divide the line joining the centers of two circles externally into.
AP : PB = (8 + x) : x
= (8 + 12) : 12 = 20 : 12 ⇒ 5 : 3
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Two circles of radii 5 cm and 3 cm touch externally, then the ratio in...
Given:
- Two circles with radii 5 cm and 3 cm.
- The circles touch externally.
- The direct common tangent divides the line joining the centers of the circles externally.
To find:
The ratio in which the direct common tangent divides the line joining the centers of the circles externally.
Solution:
Let's label the larger circle as O1 and the smaller circle as O2. The centers of the circles are labeled as C1 and C2, respectively.
Step 1: Draw the diagram
Draw the two circles O1 and O2 with radii 5 cm and 3 cm, respectively. Label the centers as C1 and C2, and draw the line joining the centers.
Step 2: Draw tangents
Draw tangents from the point of contact of the two circles to the line joining the centers. Let the points of contact be A and B, with A on circle O1 and B on circle O2.
Step 3: Join points A, B, C1, and C2
Join points A and B with the centers C1 and C2.
Step 4: Identify the triangles
We have two triangles: triangle C1AB and triangle C2AB.
Step 5: Identify the ratios
Since the triangles are similar (by the tangent-tangent theorem), the ratio of the sides is equal to the ratio of the corresponding radii.
In triangle C1AB:
- CA is the radius of circle O1 (5 cm)
- CB is the radius of circle O2 (3 cm)
- AB is the direct common tangent
In triangle C2AB:
- CA is the radius of circle O1 (5 cm)
- CB is the radius of circle O2 (3 cm)
- AB is the direct common tangent
Therefore, the ratio of the sides CA:CB in both triangles is 5:3.
Step 6: Simplify the ratios
We can simplify the ratio 5:3 by dividing both sides by 3:
CA:CB = 5/3:3/3
CA:CB = 5/3:1
Step 7: Determine the ratio in which the tangent divides the line joining the centers
The ratio in which the direct common tangent divides the line joining the centers is equal to the ratio of the corresponding sides in the triangles C1AB and C2AB.
Therefore, the ratio in which the tangent divides the line joining the centers is 5:3.
Answer:
The correct answer is option D) 5:3.
- Two circles with radii 5 cm and 3 cm.
- The circles touch externally.
- The direct common tangent divides the line joining the centers of the circles externally.
To find:
The ratio in which the direct common tangent divides the line joining the centers of the circles externally.
Solution:
Let's label the larger circle as O1 and the smaller circle as O2. The centers of the circles are labeled as C1 and C2, respectively.
Step 1: Draw the diagram
Draw the two circles O1 and O2 with radii 5 cm and 3 cm, respectively. Label the centers as C1 and C2, and draw the line joining the centers.
Step 2: Draw tangents
Draw tangents from the point of contact of the two circles to the line joining the centers. Let the points of contact be A and B, with A on circle O1 and B on circle O2.
Step 3: Join points A, B, C1, and C2
Join points A and B with the centers C1 and C2.
Step 4: Identify the triangles
We have two triangles: triangle C1AB and triangle C2AB.
Step 5: Identify the ratios
Since the triangles are similar (by the tangent-tangent theorem), the ratio of the sides is equal to the ratio of the corresponding radii.
In triangle C1AB:
- CA is the radius of circle O1 (5 cm)
- CB is the radius of circle O2 (3 cm)
- AB is the direct common tangent
In triangle C2AB:
- CA is the radius of circle O1 (5 cm)
- CB is the radius of circle O2 (3 cm)
- AB is the direct common tangent
Therefore, the ratio of the sides CA:CB in both triangles is 5:3.
Step 6: Simplify the ratios
We can simplify the ratio 5:3 by dividing both sides by 3:
CA:CB = 5/3:3/3
CA:CB = 5/3:1
Step 7: Determine the ratio in which the tangent divides the line joining the centers
The ratio in which the direct common tangent divides the line joining the centers is equal to the ratio of the corresponding sides in the triangles C1AB and C2AB.
Therefore, the ratio in which the tangent divides the line joining the centers is 5:3.
Answer:
The correct answer is option D) 5:3.
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Two circles of radii 5 cm and 3 cm touch externally, then the ratio in which the direct common tangent to the circles divides externally the line joining the centres of the circles is: (SSC CHSL 2015)a)2.5 : 1.5b)1.5 : 2.5c)3 : 5d)5 : 3Correct answer is option 'D'. Can you explain this answer?
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