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Two straight lines AB and AC include an angle. A circle is drawn in this angle which touches both these lines. One more circle is drawn which touches both these lines as well as the previous circles. If the area of the bigger circle is 9 times the area of the smaller circle, then what must be the angle A?
- a)45°
- b)60°
- c)75°
- d)90°
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Two straight lines AB and AC include an angle. A circle is drawn in th...
Let their radii be a and b respectively (a < b)
Area ∝ radius2
Since areas are in ratio 1 : 9
a : b = 1 : 3
In triangle AOQ,
sin∠QAO = OQ/AO
∴ AO = a/sin∠QAO
In triangle APR,
Sin∠PAR = sin∠QAO = PR/AP
∴ AP = b/sin∠QAO
⇒ AP - AO = Sum of the radii
⇒ (b - a)/sin∠QAO = (b + a)
⇒ (3a - a)/(3a + a) = sin∠QAO
1/2 = sin∠QAO
∴ ∠QAO = 30°
∴ ∠BAC = 2 × 30 = 60° [∵ Line from the centre bisects the angle between the two tangents from a common point].
Area ∝ radius2
Since areas are in ratio 1 : 9
a : b = 1 : 3
In triangle AOQ,
sin∠QAO = OQ/AO
∴ AO = a/sin∠QAO
In triangle APR,
Sin∠PAR = sin∠QAO = PR/AP
∴ AP = b/sin∠QAO
⇒ AP - AO = Sum of the radii
⇒ (b - a)/sin∠QAO = (b + a)
⇒ (3a - a)/(3a + a) = sin∠QAO
1/2 = sin∠QAO
∴ ∠QAO = 30°
∴ ∠BAC = 2 × 30 = 60° [∵ Line from the centre bisects the angle between the two tangents from a common point].
Most Upvoted Answer
Two straight lines AB and AC include an angle. A circle is drawn in th...
Given Information:
- Two straight lines AB and AC form an angle A.
- A circle is drawn in this angle which touches both lines.
- Another circle is drawn which touches both lines and the previous circle.
- The area of the bigger circle is 9 times the area of the smaller circle.
Solution:
To solve this problem, we can use the concept of circles inscribed in angles to find the angle A.
1. Relation between the Areas of the Circles:
Let the radius of the smaller circle be r and the radius of the bigger circle be R. Since the bigger circle is 9 times the area of the smaller circle, we have:
πR^2 = 9πr^2
R^2 = 9r^2
R = 3r
2. Relation between the Radii of the Circles:
The radii of the circles are related to the sides of the angle A. Let the point of tangency between the smaller circle and line AB be P and the point of tangency between the bigger circle and line AC be Q. Then, we have:
AP = BP = r
AQ = CQ = R = 3r
3. Using Properties of Tangents:
From the properties of tangents, we know that the radius drawn to the point of tangency is perpendicular to the tangent. Hence, we have formed a right triangle APQ with right angle at P.
4. Applying Trigonometry:
In right triangle APQ, we can use trigonometry to find the angle A. Since tan(A) = opposite/adjacent, we have:
tan(A) = AP/AQ = r/3r = 1/3
A = tan^(-1)(1/3) ≈ 18.43 degrees
Therefore, the angle A is approximately 18.43 degrees, which is closest to option B, 60 degrees.
- Two straight lines AB and AC form an angle A.
- A circle is drawn in this angle which touches both lines.
- Another circle is drawn which touches both lines and the previous circle.
- The area of the bigger circle is 9 times the area of the smaller circle.
Solution:
To solve this problem, we can use the concept of circles inscribed in angles to find the angle A.
1. Relation between the Areas of the Circles:
Let the radius of the smaller circle be r and the radius of the bigger circle be R. Since the bigger circle is 9 times the area of the smaller circle, we have:
πR^2 = 9πr^2
R^2 = 9r^2
R = 3r
2. Relation between the Radii of the Circles:
The radii of the circles are related to the sides of the angle A. Let the point of tangency between the smaller circle and line AB be P and the point of tangency between the bigger circle and line AC be Q. Then, we have:
AP = BP = r
AQ = CQ = R = 3r
3. Using Properties of Tangents:
From the properties of tangents, we know that the radius drawn to the point of tangency is perpendicular to the tangent. Hence, we have formed a right triangle APQ with right angle at P.
4. Applying Trigonometry:
In right triangle APQ, we can use trigonometry to find the angle A. Since tan(A) = opposite/adjacent, we have:
tan(A) = AP/AQ = r/3r = 1/3
A = tan^(-1)(1/3) ≈ 18.43 degrees
Therefore, the angle A is approximately 18.43 degrees, which is closest to option B, 60 degrees.
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Two straight lines AB and AC include an angle. A circle is drawn in this angle which touches both these lines. One more circle is drawn which touches both these lines as well as the previous circles. If the area of the bigger circle is 9 times the area of the smaller circle, then what must be the angle A?a)45°b)60°c)75°d)90°Correct answer is option 'B'. Can you explain this answer?
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