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Two circles with radii 5 cm and 8 cm touch each other externally at a point A. If a straight line through the point A cuts the circles at points P and Q respectively, then AP : AQ is (SSC CGL 1st Sit. 2012)
- a)8 : 5
- b)5 : 8
- c)3 : 4
- d)4 : 5
Correct answer is option 'B'. Can you explain this answer?
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Two circles with radii 5 cm and 8 cm touch each other externally at a ...
Given:
- Two circles with radii 5 cm and 8 cm
- The circles touch each other externally at point A
- A straight line through point A cuts the circles at points P and Q
To find:
The ratio of AP to AQ
Solution:
Let's consider the given information step by step.
Tangent at point A:
- The line passing through point A is the tangent to both circles at point A.
- The tangent to a circle is perpendicular to the radius at the point of contact.
- Therefore, the tangent at point A is perpendicular to the radii of both circles at point A.
Triangles formed:
- Let's consider the triangle APQ.
- Given that AP and AQ are tangents to the circles, they are perpendicular to the radii at points P and Q respectively.
- Therefore, triangle APQ is a right-angled triangle at point A.
Lengths of the sides:
- Let's consider the radii of the circles.
- The radius of the first circle is 5 cm.
- The radius of the second circle is 8 cm.
- The lengths of the sides of the triangle APQ are AP, AQ, and PQ.
Using Pythagoras theorem:
- In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Therefore, using Pythagoras theorem, we can find the relation between the sides of the triangle APQ.
Applying Pythagoras theorem:
- AP^2 + PQ^2 = AQ^2
- (5)^2 + PQ^2 = (8)^2
- 25 + PQ^2 = 64
- PQ^2 = 64 - 25
- PQ^2 = 39
- PQ = √39
Ratio of AP to AQ:
- We need to find the ratio of AP to AQ.
- Dividing both sides of the equation AP^2 + PQ^2 = AQ^2 by AP^2, we get:
- 1 + (PQ/AP)^2 = (AQ/AP)^2
- (AQ/AP)^2 = 1 + (PQ/AP)^2
- (AQ/AP) = √(1 + (PQ/AP)^2)
- Since PQ/AP = √39/5 (as PQ = √39 and AP = 5),
- (AQ/AP) = √(1 + (√39/5)^2)
- (AQ/AP) = √(1 + 39/25)
- (AQ/AP) = √(64/25)
- (AQ/AP) = 8/5
Hence, the ratio of AP to AQ is 8 : 5. Therefore, the correct answer is option 'B'.
- Two circles with radii 5 cm and 8 cm
- The circles touch each other externally at point A
- A straight line through point A cuts the circles at points P and Q
To find:
The ratio of AP to AQ
Solution:
Let's consider the given information step by step.
Tangent at point A:
- The line passing through point A is the tangent to both circles at point A.
- The tangent to a circle is perpendicular to the radius at the point of contact.
- Therefore, the tangent at point A is perpendicular to the radii of both circles at point A.
Triangles formed:
- Let's consider the triangle APQ.
- Given that AP and AQ are tangents to the circles, they are perpendicular to the radii at points P and Q respectively.
- Therefore, triangle APQ is a right-angled triangle at point A.
Lengths of the sides:
- Let's consider the radii of the circles.
- The radius of the first circle is 5 cm.
- The radius of the second circle is 8 cm.
- The lengths of the sides of the triangle APQ are AP, AQ, and PQ.
Using Pythagoras theorem:
- In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Therefore, using Pythagoras theorem, we can find the relation between the sides of the triangle APQ.
Applying Pythagoras theorem:
- AP^2 + PQ^2 = AQ^2
- (5)^2 + PQ^2 = (8)^2
- 25 + PQ^2 = 64
- PQ^2 = 64 - 25
- PQ^2 = 39
- PQ = √39
Ratio of AP to AQ:
- We need to find the ratio of AP to AQ.
- Dividing both sides of the equation AP^2 + PQ^2 = AQ^2 by AP^2, we get:
- 1 + (PQ/AP)^2 = (AQ/AP)^2
- (AQ/AP)^2 = 1 + (PQ/AP)^2
- (AQ/AP) = √(1 + (PQ/AP)^2)
- Since PQ/AP = √39/5 (as PQ = √39 and AP = 5),
- (AQ/AP) = √(1 + (√39/5)^2)
- (AQ/AP) = √(1 + 39/25)
- (AQ/AP) = √(64/25)
- (AQ/AP) = 8/5
Hence, the ratio of AP to AQ is 8 : 5. Therefore, the correct answer is option 'B'.
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Community Answer
Two circles with radii 5 cm and 8 cm touch each other externally at a ...
∴ AP : AQ = 5 : 8
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Two circles with radii 5 cm and 8 cm touch each other externally at a point A. If a straight line through the point A cuts the circles at points P and Q respectively, then AP : AQ is (SSC CGL 1st Sit. 2012)a)8 : 5b)5 : 8c)3 : 4d)4 : 5Correct answer is option 'B'. Can you explain this answer?
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