AP EAMCET Engineering 21 Sep 2020 Shift 2 Paper

(d) Since, α and β are roots of quadratic equation px2+qx+r=0,p≠0 and p,q,r are in AP such that
‌

1

α

+‌

1

β

=4‌‌......(i)
Let the p=q−d and r=q+d, where d is the common difference of AP
∴ Sum of the roots α+β=‌

−q

q−d

and product of the roots αβ=‌

q+d

q−d

∵ from Eq. (i), we have
‌

α+β

αβ

=4⇒‌

−q

q+d

=4⇒5q+4d=0‌‌.....(ii)
Now, |α−β|=√(α+β)2−4αβ
=√‌

q2

(q−d)2

−4‌

q+d

q−d

=√‌

q2−4(q2−d2)

(q−d)2

=√‌

4d2−3q2

(q−d)2

=√‌

4(− 5 4 q)2−3q2

(q+‌ 5 4 q)2

=√‌

‌ 25 4 −3

(‌ 9 4 )2

=‌

2

9

√13