Two circles touch each other internally. Prove that the tangent drawn to the circle from any point on the common tangent are equal in length.

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Solution

From A, AB and AC are are two tangents to the circle with center C
Therefore, AB = AO.....(equation ii)
also From A, AC and AO are are two tangents to the small circle.
Therefore, AC = AO...(equation ii)

From Equation i and ii
We have AB = AC
Hence proved!!

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Q. Two circles touch each other externally at point P. Q is a point on the common tangent through P. Prove that the tangents QA and QB are equal.

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

What is the maximum number of common tangents that can be drawn to two circles of different radii, touching each other internally?

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Two circles touch each other internally. Prove that the tangent drawn to the circle from any point on the common tangent are equal in length.

From A, AB and AC are are two tangents to the circle with center CTherefore, AB = AO.....(equation ii)also From A, AC and AO are are two tangents to the small circle.Therefore, AC = AO...(equation ii)From Equation i and iiWe have AB = ACHence proved!!

From A, AB and AC are are two tangents to the circle with center C
Therefore, AB = AO.....(equation ii)
also From A, AC and AO are are two tangents to the small circle.
Therefore, AC = AO...(equation ii)

From Equation i and ii
We have AB = AC
Hence proved!!