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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