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High School Mathematics - 2
8.13 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
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 = TR

Q 2: 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.

Q 3: The diagonals of a parallelogram ABCD intersect at a point E. Show that the circumcircles of triangle ABE and BCE intersect at E.

Q 4: Find the locus of the centre of circle of constant radius r, which touches a given circle of radius r1 externally.

Q 5: 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.

Q 6: Find the locus of the centre of circle of constant radius r, which touches a given circle of radius r1 internally.

Q 7: 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.

Q 8: Two circles with centres O and Ol touch externally at point A . A line is drawn to intersect these circles at B and C. Prove that the tangents at B and C are parallel.


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