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

$AP=\frac{PB}{2},CP=PD,FP=\frac{EP}{3}$.

• A. All are of equal length
• B. AB
• C. CD
• D. EF

Answer 1

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 $\Delta OPQ$ 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