Question

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

Answer 1

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

Answer 2

Answer:

Step-by-step explanation:

Let the two circles intersect at points X and Y.

So, XY is the common chord.

Suppose ‘A’ is a point on the common chord and AM and AN be the tangents drawn A to the circle

Then it’s required to prove that AM = AN.

In order to prove the above relation, following property has to be used.

“Let PT be a tangent to the circle from an external point P and a secant to the circle through P intersecting the circle at points A and B, then PT2 = PA × PB”

Now AM is the tangent and AXY is a secant

∴ AM2 = AX × AY … (i)

Similarly, AN is a tangent and AXY is a secant∴ AN2 = AX × AY …. (ii)

From (i) & (ii), we have AM2 = AN2∴

AM = AN

Therefore, tangents drawn from any point on the common chord of two intersecting circles are equal.

Hence Proved