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Q.
The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then
$ ax ^2+2 bx + c =0 $
$ \Rightarrow ax ^2+2 \sqrt{ ac x}+ c =0\left(\because b ^2= ac \right) $
$\Rightarrow( x \sqrt{ a }+\sqrt{ c })^2=0 $
$ x ^2-\frac{\sqrt{ c }}{\sqrt{ a }} \ldots \ldots(1) $
$ \text { Now, } d x ^2+2 ex + f =0 $
$ \Rightarrow d \left(\frac{ c }{ a }\right)+2 e \left[-\frac{\sqrt{ c }}{\sqrt{ a }}\right]+ f =0 $
$ \Rightarrow \frac{d c}{ a }+ f =2 e \sqrt{\frac{ c }{ a }}$
$ \Rightarrow \frac{ d }{ a }+\frac{ f }{ c }=2 e \sqrt{\frac{1}{ ac }} $
$ \Rightarrow \frac{ d }{ a }+\frac{ f }{ c }=\frac{2 e }{ b }[\text { as } b =\sqrt{ ae }]$
$ \therefore \frac{ d }{ a }, \frac{ e }{ b }, \frac{ f }{ c } \text { are in A.P. }$