Question

if from any point on common chord of two intersecting circles, tangents be drawn to the circles, then prove that they are equal.

Answer 1

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.

AM is the tangent and AXY is a secant.

∴ AM2 = AX × AY .......................(1) [ From the theorem]

AN is the tangent and AXY is a secant.

∴ AN2 = AX × AY ..................(2) [ From the theorem]

From (1) and (2), we get

AM2 = AN2

∴ AM = AN

Answer 2

Consider 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 PT2 = PA × PB .

This property is used to solve the given question.

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.

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 get

AM2 = AN2

∴ AM = AN