PROVE THAT THE PERPENDICULAR AT THE POINT OF CONTACT TO THE TANGENT TO A CIRCLE PASSES THROUGH THE CENTER.
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Given a circle with center O and AB the tangent intersecting circle at point P
and prove that OP⊥AB
We know that tangent of the circle is perpendicular to radius at points of contact Hence
OP⊥AB
So, ∠OPB=90o..........(i)
Now lets assume some point X
Such that XP⊥AN
Hence ∠XPB=90o.........(ii)
From eq (i) & (ii)
∠OPB=∠XPB=90o
Which is possible only if line XP passes though O
Hence perpendicular to tangent passes though centre
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