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Question

In the given figure two circles touch each other internally in a point A. The radius of the smaller circle with centre M is 5.
The smaller circle passes through the centre N of the larger circle. The tangent to the smaller circle drawn through C intersects the larger circle in point D. Find CD.

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Solution


CA is a secant and CP is a tangent to the smaller circle .
By secant-tangent property, we have:
CN×CA=CP210×20=CP2CP= 102 units

Now, we have:CPM=90° (Radius is perpendicular to the tangent)ADC=90° (Angle in a semicircle is a right angle)CPM=ADC Since they are corresponding angles and are equal, PM DA. In CPM andCDA, we have:CPM=CDA=90° MCP=ACD (Common)CPM ~CDA (BY AA similarity)i.e., CPCD=CMCACD=CP×CACM=102×2015=4032 units

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