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Question

Of all chords which passes through a given point inside the circle the shortest chord will always be the one with the given point as its midpoint


A

True

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B

False

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Solution

The correct option is A

True


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 OPQ 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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