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Two Circles Intersect in Points P and Q. a Secant Passing Through P Intersects the Circles in Aand B Respectively. Tangents to the Circles at a and B Intersect at T. A, Q, B and T Lie on a Circle. - ICSE Class 10 - Mathematics

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Question

Two circles intersect in points P and Q. A secant passing through P intersects the circles in Aand B respectively. Tangents to the circles at A and B intersect at T. Prove that A, Q, B and T lie on a circle.

Solution

Join PQ.
AT is tangent and AP is a chord.
∴ `∠`TAP = `∠`AQP(angles in alternate segments) ........(i)
Similarly, `∠`TBP =  `∠`BQP .......(ii)
Adding (i) and (ii)
`∠`TAP + `∠`TBP = `∠`AQP  + `∠`BQP
⇒ `∠`TAP + `∠`TBP  =  `∠`AQB ………(iii)
Now in ΔTAB,
`∠`ATB + `∠`TAP  + `∠`TBP = 180°
⇒  `∠`ATB + `∠`AQB = 180°
Therefore, AQBT is a cyclic quadrilateral.
Hence, A, Q, B and T lie on a circle.

 

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Solution Two Circles Intersect in Points P and Q. a Secant Passing Through P Intersects the Circles in Aand B Respectively. Tangents to the Circles at a and B Intersect at T. A, Q, B and T Lie on a Circle. Concept: Cyclic Properties.
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