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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
BAQ = ACB and PAB = ADB
Construction: Join OB


Directions: Solve the following
Q 1: 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.

Answer:

Q 2: 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 3: Find the locus of the centre of circle of constant radius r, which touches a given circle of radius r1 externally.
Answer:

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

Q 5: 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 6: 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 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 = TR
Answer:

Q 8: 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:

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Question 10: This question is available to subscribers only!


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