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Geometry Common Tangents To Two Circles
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Common Tangents To Two Circles

Theorem:
If two circles touch each other (internally or externally) the point of contact lies on the line through the centres.
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Example: Find the locus of the centre of a circle of constant radius (r) which touches a given circle of radius r1
(i) externally, (ii) internally, given r1 > r.
Solution: (i) Let O be the centre of the given circle of radius r1, Let O' be the centre of circle of radius r which touches the given circle externally. Then,

OO' = r1+r
O' moves in such a way that its distance from 0{a fixed point) always remains constant equal to r1+ r.
Hence, the locus of C is a circle with centre O and radius r + r1.
(ii) Let O be the centre and r1 be the radius of the given circle. Let O' be centre and r be the radius of the variable circle which touches the given circle internally.
Then,
OO' = r1 - r
O' moves in such a way that OO' remains constant equal to (r1 - r).
Hence, the locus of O' is a circle with centre O and radius r1 - r.
 
   
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