P is point inside a circle with centre O. The following conditions are given about the chords passing through P. Find the shortest chord
The shortest chord among a set of chords passing through a given point P would be the one which has its midpoint at P. Here in this case its CD since its given as
CP = PD.
Consider any other chord passing through P which doesn't have its midpoint at P, say XY let the length of CD, the chord with P as midpoint as l and d as the perpendicular length from O to CD.
Now draw a perpendicular from O to the chord XY, OQ
If you consider ΔOPQ we get that its right angled at Q, with OP as hypotenuse and OQ as altitude.
This shows that OQ<OP
So the shortest distance from centre to XY is less than shortest distance from centre to CD.
⇒ length of XY > length of CD
⇒ CD is the shortest chord possible