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CBSE Class 10 Answered

if PT is a tangent and PAB is a secant to a circle from an external point P,then show that PTsquare = PA*PB
Asked by | 23 Feb, 2013, 06:04: AM
answered-by-expert Expert Answer
Given : PAB is secant intersecting the circle with centre O at A and B and a tangent PT at T.
 
Draw OM perpendicular to AB and join OA, OB and OP
 
Since, OM is perpendicular to AB and a line drawn through the center is perpendicular to the chord, it bisects it also. (Can be proven easily by RHS congruency)
 
Therefore, AM = MB. 
 
Now, PA*PB = (PM-AM) * (PM+MB)
Since AM = MB
Therefore, PA*PB = (PM-AM) * (PM+AM)
PA*PB = PM2-AM2 (1)
 
Also, since triangle (PMO) is a right angled triangle, therefore, by pythagoras theorem, 
PM2 = OP2 - OM2
Also, triangle AMO is right angles triangle, therefore, AM2 = OA2 - OM2
 
Substituing it back in (1)
 
PA*PB = OP2 - OM2 - (OA2 - OM2)
PA*PB = OP2 - OA2
PA*PB = OP2 - OT2 (OA = OT as it is the radius of the circle)
PA*PB = PT2 (By using pythagoras theorm in triangle POT)
 
Hence, proved
Answered by | 23 Feb, 2013, 08:34: PM

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