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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: Find the locus of the centre of circle of constant radius r, which touches a given circle of radius r1 externally.Answer: Q 2: The diagonals of a parallelogram ABCD intersect at a point E. Show that the circumcircles of triangle ABE and BCE intersect at E.Answer: Q 3: Two rays ABP and ACQ are intersected by two parallel lines in B, C, P and Q respectively.Prove that the circumcircles of triangles ABC and APQ touch each other at A.Answer: Q 4: A circle with centre O intersects another circle with centre Ol in A and A and B passes through O. Tangent CD is drawn to the circle with sentre Ol. prove that OA is the bisector of angle CAB.Answer: Q 5: 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: Q 6: Two circles intersect at points A and B. From a point P on a circle two line segments PAC and PDB are drawn intersecting the other circle at points C and D. Prove that CD is parallel to tangent at P.Answer: Q 7: 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 8: Two circles touch each other at points A and B. At A, tangents AP and AQ are drawn which intersect other circles at the points P and Q respectively. Prove that AB is the bisector of angle PBQ.Answer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!