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.

8 The radius of the incircle of a triangle is 4 cm and the segments into which one side is divided by the point of contact are 6 cm and
8 cm. Determine the other two sides of the triangle.
(Hint: Equate the areas of the triangle found by using the formula √[s(s-a)(s-b)(s-c)] and also found by dividing it into three triangles.) 9 The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle. BD is a tangent to the smaller circle touching it at D. Find the length AD in figure. [Hint: Let line BD intersect the bigger circle at E. Join AE. AE = 2 × 8= 16 cm.
DE = BD = √(169 - 64) = � and ∠AED= 90°.]
   11 In figure, two circles with centres O, O' touch externally at a point A. A line through A is drawn to intersect these circles at B and C. Prove that the tangents at B and C are parallel.  [Hint: Prove that ∠ OBA = ∠ O' CA]
         12 In figure two circles intersect at two points A and B. From a point P on a circle, two line segments PAC and PBD are drawn intersecting the other circle at the points C and D. Prove that CD is parallel to the tangent at P.