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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
Sequences and Series
Solution:
$ D=q^2-4 p r$ also $q=\frac{2 p r}{p+r}$
$=\left(\frac{2 pr }{ p + r }\right)^2-4 pr =-4 pr \left[1-\frac{ pr }{( p + r )^2}\right] \Rightarrow-4 pr \left[\frac{ p ^2+ r ^2+ pr }{( p + r )^2}\right] $
$\therefore D <0 \text { roots are imaginary }$