If the ratio of roots of $a_{1} x^{2} + b_{1} x + c_{1} = 0$ be equal to the ratio of the roots of $a_{2} x^{2} + b_{2} x + c_{2} = 0,$ then $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}}$ are in

A . P .

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G . P .

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H . P .

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None of these

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Solution

The correct option is B $G . P .$
Let the ratio of the roots be $k .$
Then, the roots of $a_{1} x^{2} + b_{1} x + c_{1} = 0$ are $α, k α$
and the roots of $a_{2} x^{2} + b_{2} x + c_{2} = 0$ are $β, k β .$
$∴ α + k α = \frac{- b_{1}}{a_{1}}$ ...(1)
$α . k α = \frac{c_{1}}{a_{1}}$ ...(2)
$β + k β = \frac{- b_{2}}{a_{2}}$ ...(3)
$β . k β = \frac{c_{2}}{a_{2}} .$ ...(4)
Dividing (1) by (3), we get
$\frac{α (1 + k)}{β (1 + k)} = \frac{b_{1} a_{2}}{a_{1} b_{2}}$ , or $\frac{α}{β} = \frac{b_{1} a_{2}}{a_{1} b_{2}}$ ...(5)
Dividing (2) by (4), we get
$\frac{k α^{2}}{k β^{2}} = \frac{c_{1} a_{1}}{a_{1} c_{2}};$ or ${(\frac{α}{β})}^{2} = \frac{c_{1} a_{2}}{a_{1} c_{2}}$
$\Rightarrow {(\frac{b_{1} a_{2}}{a_{1} b_{2}})}^{2} = \frac{c_{1} a_{2}}{a_{1} c_{2}}$ (Using (5))
$\Rightarrow {(\frac{b_{1}}{b_{2}})}^{2} = \frac{c_{2} a_{2}}{a_{1} c_{2}} \times \frac{a_{1}^{2}}{a_{2}^{2}} = \frac{c_{1} a_{1}}{c_{2} a_{2}} = \frac{a_{1}}{a_{2}} . \frac{c_{1}}{c_{2}}$
$∴ \frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}}$ are in $G . P .$

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Similar questions

Q. The ratio of the roots of the

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

be equal to the ratio of roots of

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

then

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

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

and

a_{3} x^{2} + b_{3} x + c_{3} = 0

has a common root, then show that
(i)

\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

(ii)

{(\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}}

Q. If the roots of

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

are

α_{1}, β_{1},

and those of

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

are

α_{2}, β_{2}

such that

α_{1} α_{2} = β_{1} β_{2} = 1

, then

Unique point is obtained for the pair of equations $a_{1}$ x + $b_{1}$ y + $c_{1}$ = 0 and $a_{2}$ x + $b_{2}$ y + $c_{2}$ = 0 if __________ .

What can you tell about the ratio of coefficients of the lines $a_{1}$ x + $b_{1}$ y + $c_{1}$ = 0 and $a_{2}$ x + $b_{2}$ y + $c_{2}$ = 0 from the graph ?

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If the ratio of roots of $a_{1} x^{2} + b_{1} x + c_{1} = 0$ be equal to the ratio of the roots of $a_{2} x^{2} + b_{2} x + c_{2} = 0,$ then $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}}$ are in