Question

In the figure $O$ is the centre of the circle. $CD$ is a chord which is not perpendicular to the diameter $AB$. $PA=9cm$ and $PB=4cm$.
a) What is $PC\times PD$?
b) Show that the length of $PC$ and $PD$ cannot be natural numbers at a time.

Answer 1

(a)$AB$ and $CD$ are two intersecting chords of the circle.
$\therefore$ Using circle chord property we can write
$PC\times PD=AP\times PB$
$PC\times PD=9\times 4=36c{m}^2$
(b) The product of $PC$ and $PD$ is $36$.
Also AB is the diametre so $PC+PD<AB=13cm$
Only combination which will satisfy both of the above conditions is $(6,6)$.
But for this combination $PC=PD$, which is possible only when $CD\perp AB$.
It is given in the question that $CD$ is not perpendicular to $AB$.
There are no two other natural numbers whose product is 36 and sum is less than 13.
Hence PC and PD can not be natural numbers.