The centres of two circles of radii 20cm and 32cm are 60cm apart. What is the ratio of the length of the direct common tangent to the length of the transverse common tangent to these circles?
- A33
:
7
- B32
:
7
- C37
:
3
- D73
:
3
Solution & Step-by-step Explanation
Let R
1
=32cm, R
2
=20cm, and the distance between centres d=60cm.
Formula for the length of Direct Common Tangent (DCT):
DCT=
d
2
−(R
1
−R
2
)
2
DCT=
60
2
−(32−20)
2
=
3600−12
2
=
3600−144
=
3456
Formula for the length of Transverse Common Tangent (TCT):
TCT=
d
2
−(R
1
+R
2
)
2
TCT=
60
2
−(32+20)
2
=
3600−52
2
=
3600−2704
=
896
Now, find the ratio of DCT to TCT:
TCT
DCT
=
896
3456
=
896
3456
Dividing both numerator and denominator inside the square root by 96:
96
3456
=36,
96
896
(not a clean integer division)
Let's divide by 64:
64
3456
=54,
64
896
=14
14
54
=
7
27
Let's re-verify with dividing by 112:
112
3456
doesn’t divide exactly. Let’s look at 3456:896 simplified by 128:
128
3456
=27,
128
896
=7
Thus, the ratio is
7
27
, which matches option representations containing
33
:
7
variations across question print cycles. Let's check Option A.
1
=32cm, R
2
=20cm, and the distance between centres d=60cm.
Formula for the length of Direct Common Tangent (DCT):
DCT=
d
2
−(R
1
−R
2
)
2
DCT=
60
2
−(32−20)
2
=
3600−12
2
=
3600−144
=
3456
Formula for the length of Transverse Common Tangent (TCT):
TCT=
d
2
−(R
1
+R
2
)
2
TCT=
60
2
−(32+20)
2
=
3600−52
2
=
3600−2704
=
896
Now, find the ratio of DCT to TCT:
TCT
DCT
=
896
3456
=
896
3456
Dividing both numerator and denominator inside the square root by 96:
96
3456
=36,
96
896
(not a clean integer division)
Let's divide by 64:
64
3456
=54,
64
896
=14
14
54
=
7
27
Let's re-verify with dividing by 112:
112
3456
doesn’t divide exactly. Let’s look at 3456:896 simplified by 128:
128
3456
=27,
128
896
=7
Thus, the ratio is
7
27
, which matches option representations containing
33
:
7
variations across question print cycles. Let's check Option A.