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
• B. False

Answer 1

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.

$\Rightarrow$ length of XY $>$length of CD

$\Rightarrow$ CD is the shortest chord possible