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Question

If $a comma blank b comma$ and $c$ are in G.P., then the equations $a x to the power of 2 end exponent plus 2 b x plus c equals 0$ and $d x to the power of 2 end exponent plus 2 e x plus f equals 0$ have a common root if $fraction numerator d over denominator a end fraction comma fraction numerator e over denominator b end fraction comma fraction numerator f over denominator c end fraction$ are in

A.P.
G.P.
H.P.
None of these

The correct answer is: A.P.

For first equation $D equals 4 b to the power of 2 end exponent minus 4 a c equals 0$ (as given $a comma blank b comma blank c$ are in G.P.)
$rightwards double arrow$ equation has equal roots which are equal to $– fraction numerator b over denominator a end fraction$ each
Thus it should also be the root of the second equation
Thus, $d open parentheses fraction numerator negative b over denominator a end fraction close parentheses to the power of 2 end exponent plus 2 e open parentheses fraction numerator negative b over denominator c end fraction close parentheses plus f equals 0$
$rightwards double arrow d fraction numerator b to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus 2 fraction numerator b e over denominator a end fraction plus f equals 0$
$rightwards double arrow d fraction numerator a c over denominator a to the power of 2 end exponent end fraction minus 2 fraction numerator b e over denominator a end fraction plus f equals 0$ (as $b to the power of 2 end exponent equals a c$ )
$rightwards double arrow fraction numerator d over denominator a end fraction plus fraction numerator f over denominator c end fraction equals 2 fraction numerator e b over denominator a c end fraction equals 2 fraction numerator e over denominator b end fraction$

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