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Question

With respect to position of two circles which of the following statements is/are correct?

1. If one circle lies completely outside the other circle,

Number of direct common tangents = 2

Number of transverse common tangents = 2

2. If two circles touch each other externally

Number of direct common tangents = 2

Number of transverse common tangents = 1

3. If two circles touch each other internally

Number of direct common tangents = 1

Number of transverse common tangents = 0

4. If two circles intersect each other at two points

Number of direct common tangents = 2

Number of transverse common tangents = 0

5. If one circle lies completely inside the other circle

Number of direct common tangents = 0

Number of transverse common tangents = 0


A

Only 1, 2, 3

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B

Only 1, 2, 5

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C

Only 1, 2, 3, 4

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D

All 1, 2, 3, 4, 5

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Solution

The correct option is D

All 1, 2, 3, 4, 5


1. If one circle lies completely outside the other circle.

Definition of direct common tangent: If with respect to a tangent both the circles lie on the same side, this tangent is called direct common tangent. Here line 1 and 2 are direct common tangent. Number of direct common tangent = 2

Definition of transverse common tangent: If with respect to a tangent both the circles lie on the opposite side, this tangent is called transverse common tangent. Here, line 3 & 4 are transverse common tangent. Number of transverse common tangent = 2

By definition of direct and transverse tangents here line (1) and (2) are direct tangent and line 3 is transverse tangent.

Number of direct common tangents = 2

Number of transverse common tangents = 1

3. If two circles touch each other internally

By definition of direct and transverse common tangents, here, line (1) is a direct common tangent.

Number of direct common tangents = 1

Number of transverse common tangents = 0

4. If two circles intersect each other at two points

Here, line (1) and (2) are direct common tangents

Number of direct common tangents = 2

Number of transverse common tangents = 0

5. If one circle lies completely inside the other circle

We can see that there No common tangent

Number of direct and transverse common tangents = 0

So, all 1, 2, 3, 4 and 5 statements are correct.

2. If two circles touch each other externally


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