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if from any point on the common chord of two intersecting circles,tangents be drawn to the circles,prove that they are equal.
Asked by | 23 Feb, 2013, 03:49: PM Expert Answer

Let the two circles intersect at points M and M. MN is the common chord.

Suppose O is a point on the common chord and OP and OQ be the tangents drawn from A to the circle.

OP is the tangent and OMN is a secant.

According to the theorem which says that if PT is a tangent to the circle from an external point P and a secant to the circle through P intersects the circle at points A and B, then  PT2 = PA × PB

Therefore, OP2 = OM*ON (1)

Similarly, OQ is the tangent and OMN is a secant for that also.
Therefore OQ2 = OM*ON (2)

Comparing 1 and 2

OP2 = OQ2
i.e OP = OQ
Hence, he length of the 2 tangents is equal. Hence proved.
Answered by | 23 Feb, 2013, 08:20: PM

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