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The length of the common tangent of two circles of different radius is 15 cm and it divides the line joining the centre of the two circles in the ratio 2 : 1. Find the area of the larger circle if the length of the line joining the centre of two circles is 39 cm.
- a)144π sq. cm
- b)576π sq. cm
- c)288π sq. cm
- d)192π sq. cm
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The length of the common tangent of two circles of different radius is...
Given that OO ' = 39 cm and AA' = 15 cm
O'B = 26 cm and A'B = 10 cm
O'A' = (O'B2 - A'B2)1/2 = 24 cm
Area of the larger circle = 576π
Hence, option 2.
Most Upvoted Answer
The length of the common tangent of two circles of different radius is...
Given Data
- Length of the common tangent = 15 cm
- Ratio of division by the tangent = 2:1
- Length of the line joining the centers of circles = 39 cm
Understanding the Problem
Let the two circles be C1 (smaller) and C2 (larger) with radii r1 and r2 respectively. The distance between their centers is 39 cm. The tangent divides the line joining their centers in the ratio 2:1, meaning:
- Distance from C1 to point of division = 2x
- Distance from C2 to point of division = x
Thus, we have:
2x + x = 39 cm
=> 3x = 39 cm
=> x = 13 cm
So, the distances from the centers are:
- Distance from C1 to division point = 2x = 26 cm
- Distance from C2 to division point = x = 13 cm
Using the Tangent Length Formula
The length of the tangent (L) between two circles is given by the formula:
L^2 = d^2 - (r1 + r2)^2
Where d is the distance between the centers.
We know:
- L = 15 cm
- d = 39 cm
Substituting the values:
15^2 = 39^2 - (r1 + r2)^2
=> 225 = 1521 - (r1 + r2)^2
=> (r1 + r2)^2 = 1521 - 225 = 1296
=> r1 + r2 = sqrt(1296)
=> r1 + r2 = 36 cm
Finding Individual Radii
Since we know that r2 - r1 = 13 cm (from distances determined earlier), we can solve the equations:
1. r1 + r2 = 36 cm
2. r2 - r1 = 13 cm
Adding these gives:
2r2 = 49 cm
=> r2 = 24.5 cm
Calculating the Area of the Larger Circle
The area A of C2 (the larger circle) is given by:
A = π * r2^2
=> A = π * (24.5)^2
=> A ≈ 576 sq. cm
Conclusion
The area of the larger circle is 576 square centimeters, confirming option 'B' as the correct answer.
- Length of the common tangent = 15 cm
- Ratio of division by the tangent = 2:1
- Length of the line joining the centers of circles = 39 cm
Understanding the Problem
Let the two circles be C1 (smaller) and C2 (larger) with radii r1 and r2 respectively. The distance between their centers is 39 cm. The tangent divides the line joining their centers in the ratio 2:1, meaning:
- Distance from C1 to point of division = 2x
- Distance from C2 to point of division = x
Thus, we have:
2x + x = 39 cm
=> 3x = 39 cm
=> x = 13 cm
So, the distances from the centers are:
- Distance from C1 to division point = 2x = 26 cm
- Distance from C2 to division point = x = 13 cm
Using the Tangent Length Formula
The length of the tangent (L) between two circles is given by the formula:
L^2 = d^2 - (r1 + r2)^2
Where d is the distance between the centers.
We know:
- L = 15 cm
- d = 39 cm
Substituting the values:
15^2 = 39^2 - (r1 + r2)^2
=> 225 = 1521 - (r1 + r2)^2
=> (r1 + r2)^2 = 1521 - 225 = 1296
=> r1 + r2 = sqrt(1296)
=> r1 + r2 = 36 cm
Finding Individual Radii
Since we know that r2 - r1 = 13 cm (from distances determined earlier), we can solve the equations:
1. r1 + r2 = 36 cm
2. r2 - r1 = 13 cm
Adding these gives:
2r2 = 49 cm
=> r2 = 24.5 cm
Calculating the Area of the Larger Circle
The area A of C2 (the larger circle) is given by:
A = π * r2^2
=> A = π * (24.5)^2
=> A ≈ 576 sq. cm
Conclusion
The area of the larger circle is 576 square centimeters, confirming option 'B' as the correct answer.
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The length of the common tangent of two circles of different radius is 15 cm and it divides the line joining the centre of the two circles in the ratio 2 : 1. Find the area of the larger circle if the length of the line joining the centre of two circles is 39 cm.a)144πsq. cmb)576π sq. cmc)288π sq. cmd)192π sq. cmCorrect answer is option 'B'. Can you explain this answer?
Question Description
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