Question

two circles with Centres at A and B and radii equal to R1 and R2 respectively,intersect each other at C and D .through C a line parallel to AB is drawn intersecting the circles at X and Y . what is the ratio of XY to AB​

Answer 1

The ratio of XY to AB​  is 2

Since the ratio will remain same for any circle, lets assume 2 circles of same radius R.

• ie, R1 = R2 =R

Let P be be foot of perpendicular from centre to the chord XC.

Now consider ΔACE and ΔPAC

• PA = CE , distance between parallel lines are same at all points.

• ∠APC = ∠ACE = 90 degree, again parallel line angles.

In quadrilateral PACE. 2 opposite sides are equal and 2 angles are 90 degree. Therefore its a rectangle.

Hence,

• PC  = AE Since APCE forms a rectangle.

• XC = 2PC = 2AE

• Also XY = 2XC = 4PC = 4AE = 2( 2AC) = 2 AB

• Therefore, XY/AB =2