Question

P is point inside a circle with centre O. The following conditions are given about the chords passing through P. Find the shortest chord AP=PB2,CP=PD,FP=EP3.

A

All are of equal length

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B

AB

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C

CD

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D

EF

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Solution

The correct option is C CD 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. Proof 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 OQ<P 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

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