Question

If the two equations ax^2 + bx + c = 0 and px^2 + qx + r = 0 have a common root, then the value of ( aq - bp ) ( br - cq ) is   
( a ) - ( ar - cp )^2
 ( b ) ( ap - cr )^2
( c ) ( ac - pr )^2
( d ) ( ar - cp )^2 

Answer 1

the answerr is the option a .

Answer 2

the answer is option d)
see
( aq - bp ) ( br - cq )=abqr-acq²-b²pr+bpcq
then sum of roots for first equation
α+β=-b/a,αβ=c/a
and second is
α+ω=-q/p,αω=r/p
so
-b/a-β=-q/p-ω
ω=b/a-q/p+β
as
c/(aβ)=r/(pω)
ω=raβ/cp=b/a-q/p+β
therefore β=(b/a-q/p)/((ra/cp)-1)
$\beta = \frac{(pb-aq)}{ap} \frac{(ra-cp)}{cp}$
and 
ω=(ra/cp)((b/a-q/p)/((ra/cp)-1))
$= \frac{ra}{cp} \frac{(pb-aq)}{ap} \frac{(ra-cp)}{cp}$