If p , q , r in H.P. and p and r be different having same sign then the roots of the equation p x2+q x+r=0 are

Question

If p , q , r in H.P. and p and r be different having same sign then the roots of the equation p x2+q x+r=0 are (A) real and equal (B) real and distinct (C) irrational (D) imaginary

Tardigrade Admin · Accepted Answer

D=q2-4 p r also q=(2 p r/p+r) =((2 pr / p + r ))2-4 pr =-4 pr [1-( pr /( p + r )2)] ⇒-4 pr [( p 2+ r 2+ pr /( p + r )2)] ∴ D <0 text roots are imaginary

Q. If $p, q, r$ in H.P. and $p$ and $r$ be different having same sign then the roots of the equation $p x^{2} + q x + r = 0$ are

real and equal

real and distinct

irrational

imaginary

$D = q^{2} - 4 p r$ also $q = \frac{2 p r}{p + r}$
$= (\frac{2 p r}{p + r})^{2} - 4 p r = - 4 p r [1 - \frac{p r}{( p + r ) ^{2}}] \Rightarrow - 4 p r [\frac{p ^{2} + r ^{2} + p r}{( p + r ) ^{2}}]$
$∴ D < 0 roots are imaginary$

Q. If p,q,r in H.P. and p and r be different having same sign then the roots of the equation px2+qx+r=0 are