Comprehension
In the adjoining figure I and II, are circles with P and Q respectively. The two circles touch each other and have a common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the ratio 4:3. It is also known that the length of PO is 28 cm.
Question: 1
What is the ratio of the length of PQ to that of QO?
What is the ratio of the length of PQ to that of QO?
Show Hint
In tangent problems involving two circles, use the ratio of diameters and distances from the center to find the length ratio.
Updated On: Aug 1, 2025
- 1 : 4
- 1 : 3
- 3 : 8
- 3 : 4
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The Correct Option is B
Solution and Explanation
The two circles touch each other externally, and the diameters of circles I and II are in the ratio 4:3.
Since the length of PO is 28 cm and the diameter ratio is 4:3, we can calculate the length of PQ and QO based on this ratio.
Thus, the ratio of the length of PQ to that of QO is \( 1 : 3 \).
\[
\boxed{1 : 3}
\]
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Question: 2
What is the radius of the circle II?
What is the radius of the circle II?
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Use the ratio of diameters and known distances to calculate the radii of the circles.
Updated On: Aug 1, 2025
- 2 cm
- 3 cm
- 4 cm
- 5 cm
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The Correct Option is B
Solution and Explanation
Given that the diameters of the two circles are in the ratio of 4:3, and the distance from P to O is 28 cm, we can calculate the radius of circle II by using the ratio.
The length of PO is 28 cm, and since the radii are in the ratio 4:3, we find the radius of circle II is 3 cm.
\[
\boxed{3 \, \text{cm}}
\]
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Question: 3
The length of SO is
The length of SO is
Show Hint
For problems involving tangents and circles, apply the Pythagorean theorem to relate the distances and radii.
Updated On: Aug 1, 2025
- \( 8\sqrt{3} \, \text{cm} \)
- \( 10\sqrt{3} \, \text{cm} \)
- \( 12\sqrt{3} \, \text{cm} \)
- \( 14\sqrt{3} \, \text{cm} \)
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The Correct Option is B
Solution and Explanation
Using geometry and the Pythagorean theorem, we can find the length of SO by relating the radius of circle II, the distance from P to O, and the angle formed by the tangent at point S. Solving gives the length of SO as \( 10\sqrt{3} \, \text{cm} \).
\[
\boxed{10\sqrt{3} \, \text{cm}}
\]
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