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

The roots of the equation x^(2)-2sqrt(3)x+3=0 are

If the equation x^(4)-4x^(3)+ax^(2)+bx+1=0 has four positive roots, fond the values of a and b.

If the equation x^(3) +px +q =0 has three real roots then show that 4p^(3)+ 27q^(2) lt 0 .

The equations |{:(7,6,x),(2,x,2),(x,3,7):}|=0 has one root x = -9 then others roots …….

If one roots of the equation x^(2)-sqrt(5)x-19=0 is (9+sqrt(5))/2 then find the other root.

The value of b for which the equation x^2+bx-1=0 and x^2+x+b=0 have one root in common is:

If b^(2)ge4ac for the equation ax^(4)+bx^(2)+c=0 then all the roots of the equation will be real if

The sum of all the real roots of the equation |x-2|^2+|x-2|-2=0 is

Find the nature of the roots of the quadratic equations. If real roots exist, find them 2x^(2)-6x+3=0