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### High School Mathematics - 28.13 Secants

 Theorem: (Tangent- Secant theorem or Alternate Segment theorem) The angle between chord AB and tangent PQ through the point of contact is equal to the angle in the alternate segment. Given: PQ is a tangent at A to the circle with O as center. AB is a chord. To prove: If C is a point on a major arc and D is a point on a minor arc with respect to the chord AB then BAQ = ACB and PAB = ADB Construction: Join OB Directions: Solve the following
 Q 1: Two circles touch externally at a point P. From a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and R respectively. Prove that TQ = TRAnswer: Q 2: Find the locus of the centre of circle of constant radius r, which touches a given circle of radius r1 externally.Answer: Q 3: Find the locus of the centre of circle of constant radius r, which touches a given circle of radius r1 internally.Answer: Q 4: Two circles with centres O and Ol touch internally at a point P. A line is drawn to pass through P intersecting the two circles at Q and R respectively. Prove that OQ ll OlR.Answer: Question 5: This question is available to subscribers only! Question 6: This question is available to subscribers only!