Question

if ratio of the roots of equation px2+qx+q=0is a:b.prove that √a÷b +√b÷a +√q÷p=0

Answer 1

Let the roots of the equation be r1,r2r1,r2. Now, we know from Vieta's formulas that,
r1+r2=−qpr1+r2=−qp ... (1)
r1r2=qpr1r2=qp ... (2)

Now, (r1r2−−√+r2r1−−√)2(r1r2+r2r1)2
=r1r2+r2r1+2=r1r2+r2r1+2
=r21+r22r1r2+2=r12+r22r1r2+2
=r21+r22+2r1r2r1r2=r12+r22+2r1r2r1r2
=(r1+r2)2r1r2=(r1+r2)2r1r2

Hence,
r1r2−−√+r2r1−−√=r1+r2r1r2√r1r2+r2r1=r1+r2r1r2

Substitute r1r2=abr1r2=ab, (1) and (2) to obtain the desired result.