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?
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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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