Q. 54.1( 540 Votes )

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Answer :

to Prove: A line drawn perpendicular from P will pass from O.
Given: APB is tangent to circle with centre O.
 
Let us consider a circle with centre O. Let AB be a tangent which touches the circle at P.


Proof: 

Assume that the perpendicular to AB at P does not pass through centre O. Let it pass through another point O'. Join OP and O'P.



As perpendicular to AB at P passes through O', therefore,


O'PB = 90° ... (1)


O is the centre of the circle and P is the point of contact.
We know the line joining the centre and the point of contact to the tangent of the circle are perpendicular to each other.


OPB = 90° ... (2)


Comparing equations (1) and (2), we obtain


O'PB = OPB ... (3)


From the figure, it can be observed that,


O'PB < OPB ... (4)


Therefore, O'PB = OPB is not possible. It is only possible, when the line O'P coincides with OP.


Therefore, the perpendicular to AB through P passes through centre O.
Hence, Proved.

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