QJEEM23Ma4479: JEE Main 2023 Mathematics - Class 11 JEEM23Ma4479 Solution: The parabolas : and intersect on the line . If are positive…

Let point of intersection be ( $\alpha,1$ )

$\alpha x^2+2b\alpha+c=0$ ..... (i)

and $d\alpha^2+2e\alpha+f=0$ .... (ii)

$\Rightarrow a\alpha^2+2\sqrt{ac}\alpha+c=0$ ( $\because$ $b^2=ac$ )

${\left( {\sqrt a \alpha + \sqrt c } \right)^2} = 0$

$\alpha = - \sqrt {{c \over a}}$

Put the value of $\alpha$ in (ii),

$d{c \over a} - 2e\sqrt {{c \over a}} + f = 0$

${d \over a} - {{2e} \over {\sqrt {ac} }} + {f \over c} = 0$

${d \over a} + {f \over c} = 2{e \over b}$

${d \over a},{e \over b},{f \over c}$ are in A.P.