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प्रश्न
Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangents at P and Q intersect at point T; show that the points P, B, Q and T are concyclic.
उत्तर
Join AB, PB and BQ
TP is the tangent and PA is a chord
∴ ∠TPA = ∠ABP ...(i) (Angles in alternate segment)
Similarly,
∠TQA = ∠ABQ ...(ii)
Adding (i) and (ii)
∠TPA + ∠TQA = ∠ABP + ∠ABQ
But, ΔPTQ,
∠TPA + ∠TQA + ∠PTQ = 180°
`=>` ∠PBQ = 180° – ∠PTQ
`=>` ∠PBQ + ∠PTQ = 180°
But they are the opposite angles of the quadrilateral
Therefore, PBQT are cyclic.
Hence, P, B, Q and T are concyclic.
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Two circles with centres O and O' touch each other at point L. Prove that, a tangent through L bisects the common tangent AB of the two circles at point M.
Given: AB is a common tangent of the two circles that touch each other at point L. ML is a tangent through point L.
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M is an external point that draws two tangents, MA and ML to the circle with the centre O.
So, `square` = `square` ......(i)
Similarly, M draws two tangents ML and MB to the circle with the centre O'.
So, `square` = `square` ......(ii)
From the equations (i) and (ii),
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Hence, the tangent at the point L, bisects the common tangent, AB of the two circles at point M.
