If two given circles are touches internally with each other, take an example to understand the concept of internally touches circles.

Consider the given circles

and

Let

and

be the centre and radius of the circle (i) respectively, now to find centre and radius compare the equation of circle with general equation of circle

to get centre and radius, we have

Centre

and

Radius

Let

and

be the centre and radius of the circle (ii) respectively, now to find centre and radius compare the equation of circle with general equation of circle

to get centre and radius, we have

Centre

and

Radius

Using the distance formula, we find the distance between the centres of the given circles, we have

Now subtracting the radius from second from the first circle, we have

This shows that the distance between the centres of given circles is equal to the difference of their radii. This is only possible if the circle touches each other internally as shown in the given diagram.

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