Home
Class 12
MATHS
If the equation x^(2)-px+q=0 and x^(2)-a...

If the equation `x^(2)-px+q=0` and `x^(2)-ax+b=0` have a comon root and the other root of the second equation is the reciprocal of the other root of the first, then prove that `(q-b)^(2)=bq(p-a)^(2)`.

Text Solution

Verified by Experts

Let `alpha` and `beta` be the roots of `x^(2)-px+q=0`.Then
`alpha+beta=p` …….i
`alpha beta=q` ……………..ii
And `alpha` and `1/(beta)` be the roots of `x^(2)-ax+b=0`. Then
`alpha+1/(beta)=a` ……….iii
`(alpha)/(beta)=b` ……..iv
Now LHS`=(q-b)^(2)`
`=(apha beta-(alpha)/(beta))^(2)` [from Eqs (ii) and(iv) ]
`=alpha^(2)(beta-1/(beta))^(2)=alpha^(2)[(alpha+beta)-(alpha+1/(beta))]^(2)`
`=alpha^(2)(p-a)^(2)` [from Eqs (i) and (iii)]
`=apha .beta . (alpha)/(beta)(p-a)^(2)`
`=pq(p-a)^(2)` [from Eqs (ii) and (iv) ]
`=`RHS
Promotional Banner

Similar Questions

Explore conceptually related problems

If the equation x^2+2x+3=0 and ax^2+bx+c=0 have a common root then a:b:c is

If the equations x^(2)+a x+b=0 and x^(2)+b x+a=0(a ne b) have a common root, then a+b=

If the equation : x^(2 ) + 2x +3=0 and ax^(2) +bx+ c=0 a,b,c in R have a common root then a: b: c is :

If one root of the equation x^(2) + px + q = 0 is square of the other root, then :

If one of the roots of the equations x^(2) - 5x=0 is zero then the other roots is :

If one root of the equation 5 x^(2)+13 x+k=0 is reciprocal of other, then the value of k is

If p and q are the roots of the equation x^2-p x+q=0 , then

If p and q are the roots of the equation x^(2)+p x+q=0 then

If one root of the equation 6x^2 − 2 x + ( λ − 5 ) = 0 be the reciprocal of the other, then λ =

If the equations : x^(2) + 2x + 3 = 0 and ax^(2) + bx + c =0 a, b,c in R, Have a common root, then a: b : c is :