UPSC CDS Math Circle Questions

We have, One trapezium ABCD.
AB∥CD and AD⟂AB
And, the trapezium has an incircle which touches AB at E and CD at F.
Here EB=25cm and FC=16cm
If we draw one line from E to F which is pass through centre of circle then EF is tangent to circle and it makes 90∘ angle with AB and CD respectively.
Then EF∥AD
We draw a line from C to EB at position N means CN‌⟂‌AB
By the rule of tangent if two tangents which touches at different position in a circle has meet at one point then length is same from the particular position to tangent.
So, here MB and EB is tangent to circle and meet at B.
BM=EB=25cm
And, FC and MC is tangent to circle and meet at C.
FC=MC=16cm
Then CB=16+25=41cm
In a ∆CNB,
412=92+CN2
⇒CN=40cm
We can see CN=FE=AD=40cm
So. diameter is 40cm.