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Two circles with radii 5cm and 8 cm touch each other externally at a point A. If a straight line through t h e point Acuts the circles at points P and Q respectively, the AP : AQ is
- 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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Most Upvoted Answer
Two circles with radii 5cm and 8 cm touch each other externally at a p...
Solution:
Given, two circles with radii 5 cm and 8 cm touch each other externally at point A.
Let O1 and O2 be the centres of the circles with radii 5cm and 8 cm respectively.
Join O1A and O2A.
Let the straight line through A intersect the circles at points P and Q respectively.
Join OP and OQ.
We need to find AP : AQ.
Let AP = x and AQ = y.
To find x and y, we need to use the property that tangents drawn to a circle from an external point are equal in length.
Hence, PA = AQ.
By Pythagoras theorem, in right triangle O1AP, we have
O1P^2 = O1A^2 + AP^2
=> O1P^2 = (5)^2 + x^2
=> O1P^2 = 25 + x^2
Similarly, in right triangle O2AQ, we have
O2Q^2 = O2A^2 + AQ^2
=> O2Q^2 = (8)^2 + y^2
=> O2Q^2 = 64 + y^2
Since PA = AQ, we have
x + y = O1P + O2Q
=> x + y = OA
=> x + y = 13
Squaring both sides, we get
x^2 + 2xy + y^2 = 169
Substituting the values of O1P^2 and O2Q^2, we get
25 + x^2 + 2xy + 64 + y^2 = 169
=> x^2 + y^2 + 2xy = 80
=> (x + y)^2 - 2xy = 80
=> 13^2 - 2xy = 80
=> 169 - 80 = 2xy
=> 2xy = 89
=> xy = 89/2
Therefore, AP : AQ = x : y = 5 : 8.
Hence, the correct option is (b) 5 : 8.
Given, two circles with radii 5 cm and 8 cm touch each other externally at point A.
Let O1 and O2 be the centres of the circles with radii 5cm and 8 cm respectively.
Join O1A and O2A.
Let the straight line through A intersect the circles at points P and Q respectively.
Join OP and OQ.
We need to find AP : AQ.
Let AP = x and AQ = y.
To find x and y, we need to use the property that tangents drawn to a circle from an external point are equal in length.
Hence, PA = AQ.
By Pythagoras theorem, in right triangle O1AP, we have
O1P^2 = O1A^2 + AP^2
=> O1P^2 = (5)^2 + x^2
=> O1P^2 = 25 + x^2
Similarly, in right triangle O2AQ, we have
O2Q^2 = O2A^2 + AQ^2
=> O2Q^2 = (8)^2 + y^2
=> O2Q^2 = 64 + y^2
Since PA = AQ, we have
x + y = O1P + O2Q
=> x + y = OA
=> x + y = 13
Squaring both sides, we get
x^2 + 2xy + y^2 = 169
Substituting the values of O1P^2 and O2Q^2, we get
25 + x^2 + 2xy + 64 + y^2 = 169
=> x^2 + y^2 + 2xy = 80
=> (x + y)^2 - 2xy = 80
=> 13^2 - 2xy = 80
=> 169 - 80 = 2xy
=> 2xy = 89
=> xy = 89/2
Therefore, AP : AQ = x : y = 5 : 8.
Hence, the correct option is (b) 5 : 8.
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Community Answer
Two circles with radii 5cm and 8 cm touch each other externally at a p...
c) 3 : 4
If two circles with radii 5cm and 8cm touch each other externally at a point A, the distance between the centers of the circles will be the sum of the radii, or 5 + 8 = 13cm.
If a straight line through the point A cuts the circles at points P and Q, the distances from the centers of the circles to points P and Q will be 3cm and 4cm, respectively.
Thus, the ratio of AP to AQ will be 3 : 4, so the correct answer is c) 3 : 4.
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Two circles with radii 5cm and 8 cm touch each other externally at a point A. If a straight line through t h e point Acuts the circles at points P and Q respectively, the AP : AQ isa)8 : 5b)5 : 8c)3 : 4d)4 : 5Correct answer is option 'B'. Can you explain this answer?
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