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Analytical Geometry

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4 Jan 2012 19:16:12 IST

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Circles- AIEEE Level Question

Mathematics

A line is drawn through a fixed point P() to cut the circle x²+ y²= r² at A and B. Then PAXPB is equal to-

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Arjun Virmani

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Joined: 1 Jun 2009

Posts: 1294

5 Jan 2012 17:11:24 IST

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PA * PB = PT².............. where PT is tangent at T (to the circle )

just make a few construction :

Draw PT (tangent to circle at T from P).....

join OT (O is center of circle) ; OT is perpendicular to PT as it is from center!!

Draw OE where OE is perpendicular to AB (or PA or PB as they are same lines).........; perpendicular to a chord from center bisects the chord........ AE = BE......

join PO..

now, by pythagorus theorem in the following triangles

in tri TOP:

PT² + r² = PO²....eq 1

in tri POE :

PO² = PE² + OE²............eq 2

in tri AOE

OE² + AE² = r² ................eq 3

on adding the equations we get!!!!!

PT²+ AE² = PE²............

PT²= PE² - AE² = (PE + AE) * ( PE - AE) = (PE + BE) * ( PE - AE) = PB * PA!!!!! [ as, AE = BE ]

PA * PB = PT²

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