XY=9,XZ=12, ZX=15 a perpendicular is drawn from Y to meet at ZX at A and a circle with Diameter AY is drawn which cuts XY and YZ at B and C respectively.what is the ratio of BX:CZ

Let XYZ be a right triangle, right angled at Y. XY = 9, XZ =12 , ZX=15
Perpendicular meets ZX at A. Now draw the required circle.
Triangle XAY is similar to triangle xyz so XA/XY = XY/XZ => XA = 9*9/15= 27/5, So AZ =15-27/5 = 48/5
Now using tangent secant theorum we have xa^2 = xb*xy This gives xb = (27/5)^2/9 => 81/25
Similarly az^2 = cz*yz so cz = (48/5)^2/12 => 192/5, So the required ratio is 192:81 => 64:27