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Polynomial Functions

lf a and ...

Question

lf $a$ and $b$ are distinct positive real numbers such that
$a, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, b$ are in A.P.,
$a, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b$ are in G.P. and
$a, c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, b$ are in H.P.,
then the roots of $a_{3} x^{2} + b_{3} x + c_{3} = 0$ are

A

real and distinct

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B

real and equal

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C

imaginary

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D

rational

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Solution

The correct option is C imaginary
We know that $(G M)^{2} = (A M) (H M)$
$\Rightarrow$ $(b_{3})^{2} = (a_{3}) (c_{3})$
So, $(b_{3})^{2} - 4 (a_{3}) (c_{3}) <= 0$
Hence, the roots are imaginary.

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Q. If each pair of equations

a_{1} x^{2} + b_{1} x + c_{1} = 0, a_{2} x^{2} + b_{2} x + c_{2} = 0

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has a common root, then show that
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\frac{c_{1} a_{2} + c_{2} a_{1}}{c_{1} a_{2} - c_{2} a_{1}} + \frac{a_{1} a_{2} b_{3}}{a_{3} (a_{1} b_{2} - a_{2} b_{1})} = 0

{(\frac{a_{1} b_{2} - a_{2} b_{1}}{a_{1} c_{2} - a_{2} c_{1}})}^{2} = \frac{a_{1} a_{2} c_{3}}{c_{1} c_{2} a_{3}}

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