Question

Show that if the ratio of roots of a1x^2+b1x+c1=0 be equal to the ratio of roots of a2x^2+b2x+c2=0,
a1/a2, b1/b2, c1/c2 are in GP

Answer 1

Hence, it is shown that if the ratio of roots of a1x^2+b1x+c1=0 be equal to the ratio of roots of a2x^2+b2x+c2=0,  then a1/a2, b1/b2, c1/c2 are in G.P.

Step-by-step explanation:

Given,

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

$a_{2} x^{2} + b_{2}x + C_{2} = 0$

Assume the ratio of these roots be k,

so,

$a_{1} x^{2} + b_{1}x + c_{1} = 0$ = α have the roots kα

& $a_{2} x^{2} + b_{2}x + C_{2} = 0$ = β have the roots kβ

∵ α + kα = (-$b_{1}$)/$a_{1}$  ...(i)

α.kα = $C_{1}$/$a_{1}$   ...(ii)

β + kβ = ($-b_{2}$)/$a_{2}$  ...(iii)

β.kβ = $C_{2}$/$a_{2}$  ...(iv)

Now, by dividing the equation (i) by (iii)

$\frac{\alpha(1 + k)}{\beta (1 + k)}$ = $b_{1} a_{2} /a_{1} b_{2}$ or α/β =

Now, by dividing the equation (ii) by (iv)

$\frac{k\alpha ^2}{k\beta ^2}$ = ($C_{1}$$a_{1}$)/$a_{1}$$C_{2}$ or (α/β)^2 = $C_{1}$$a_{2}$/$a_{1}$$C_{2}$

By using the equation (v),

⇒ ($b_{1} a_{2} /a_{1} b_{2}$ )^2 =  $C_{1}$$a_{2}$/$a_{1}$$C_{2}$                 (through equation (v))

⇒ $(b_{1} /b_{2})^2 = C_{2} a_{2}/ a_{1} C_{2}/ * a_{1}^{2} /a_{2}^2$ = $a_{1} /a_{2}. C_{1} /C_{2}$

Thus, $a_{1} /a_{2}), b_{1} /b_{2}, c_{1} /c_{2}$ are in G.P.

Learn more: Geometric Expression

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