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-

Howu?? :P

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