Question:
If from any point on the common chord of two interesting circle, tangents be drawn to the circles, prove that they are equal.
Solution:
Let the two circles intersect at points X and Y. XY is the common chord.
Suppose A is a point on the common chord and AM and AN be the tangents drawn from A to the circle.
We need to show that AM = AN.
In order to prove the above relation, following property will be used.
“Let PT be 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 PT 2 = PA × PB”.
Now, AM is the tangent and AXY is a secant.
∴ AM2 = AX × AY ...(1)
AN is the tangent and AXY is a secant.
∴ AN2 = AX × AY ...(2)
From (1) and (2), we have
AM2 = AN2
∴ AM = AN