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# In the Given Figure, Mn is the Common Chord of Two Intersecting Circles and Ab is Their Common Tangent. Prove that the Line Nm Produced Bisects Ab at P. - Mathematics

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#### Question

In the given figure, MN is the common chord of two intersecting circles and AB is their common tangent.

Prove that the line NM produced bisects AB at P.

#### Solution

From P, AP is the tangent and PMN is the secant for first circle.
∴ AP^2 = PM × PN …… (i)
Again from P, PB is the tangent and PMN is the secant for second circle.
∴ PB^2 = PM × PN ……..(ii)
From (i) and (ii)
AP^2 =  PB^2
⇒ AP = PB
Therefore, P is the midpoint of AB.

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