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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 23rd February 2013, 3:19 PM
Answered by Expert

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 Expert 23rd February 2013, 8:20 PM
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