wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

If α,β are the real and distinct roots of x2+px+q=0 and α4,β4 are the roots of x2rx+5=0, then the equation x24qx+2a2r=0 always has:


A

two real roots

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B

one positive root and one negative root

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C

two positive roots

No worries! We‘ve got your back. Try BYJU‘S free classes today!
D

two negative roots

No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is B

one positive root and one negative root


α,β are the roots of x2+px+q=0
α+β=p and αβ=q
α2+β2=(α+β)22αβ=p22q
α4+β4=(α2+β2)22α2β2
=(p22q)22q2
α4 and β4 are the roots of x2rx+s=0
α4β4=s
α4+β4=r

we Know α4+β4=(p22q)22q2
(p22q)22q2=r .............(1)
The roots of the equation x24qx+2q2r=0 are

x=4q±(4q)24(2q2r)2
x=4q±24q22q2+r2
x=2q±2q2+r

From (1), 2q2+r=(p22q)2
Roots =2q±(p22q)2
=2q±(p22q)
One of the roots = 2q+(p22q)
x=p2, which is +ve
Second root is 2q(p22q)
x=4qp2
p24q>0, since the first quadratic has two real roots 4qp2 is negative

Method 2–––––––––
For the equation x2+px+q,α,β are the roots
(α+β)=p and αβ=q
α4+β4=(α+β)22αβ=p22q
α4+β4=(α2+β2)22α2β2=p22q)22q2 ...........(1)

Now, for the equation x2rx+s=0,α4,β4 are the roots.
α4β4=s and α4+β4=r .........(2)
From(1), (2)
r=(p22q)22q2 ..........(3)
Consider the equation x24ax+2q2r=0,
Product of the roots =2q2r
=2q2[(p22q)22q2] (using(3))
=2q2[p4+4q24qp22q2]
=p4+4qp2

(p24q>0 [ 1st equation has real roots ]
Product of roots = -ve
Roots are of opposite sign, if they are real
Consider discriminant of the equation

x24ax+2q2r=0
=(4q2)4(2q2r)
=16q24(2q2r)
=4(4q22q2+r)
=4(2q2+r)

r+2q2=(p22q)2(from(3))
is +ve
So, the roots are real and of opposite sign


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Algebra of Roots of Quadratic Equations
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon