If this is equal to then .
But the question says that there must be be only ONE single condition , not three . But from the three given by you...it can be converted into two...which are.."Since and , ." and . The answer given at the back of the book is . But the can't simply be ignored , can it ?
I've got it !
If be a factor of then it must vanish for .
Putting in , we get , or .
Similarly if be a factor of then it must vanish for .
Putting in , we get or .
Then since satisfies both the equations
.....................
and, .......................
The condition which is required for both the above equations to have both the roots common is actually the required condition .
On taking some determinants we easily get that, the required condition is
Q.E.D