If - is common root of both these equations ax2-bx-c-0 bx2-cx-a-0-then - 2ab-c2-cb-a2-1ac-b2

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Standard XII

Mathematics

Nth Term of GP

If α is com...

Question

If $α$ is common root of both these equations
$a x^{2} + b x + c = 0$
$b x^{2} + c x + a = 0$ ,then
$\frac{α^{2}}{a b - c^{2}} = \frac{α}{c b - a^{2}} = \frac{1}{a c - b^{2}}$

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Solution

Let $α$ be the common root of the given equations $a x^{2} + b x + c = 0$ and $b x^{2} + c x + a = 0$
Then $a α^{2} + b α + c = 0$ and $b α^{2} + c α + a = 0$
Eliminating $α$ as shown below(in the diagram part )we get
$\frac{α^{2}}{b a - c^{2}} = \frac{α}{c b - a^{2}} = \frac{1}{a c - b^{2}}$

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If α is common root of both these equationsax2+bx+c=0 bx2+cx+a=0,thenα2ab−c2=αcb−a2=1ac−b2

Let α be the common root of the given equations ax2+bx+c=0 and bx2+cx+a=0Then aα2+bα+c=0 and bα2+cα+a=0Eliminating α as shown below(in the diagram part )we getα2ba−c2=αcb−a2=1ac−b2

If $α$ is common root of both these equations
$a x^{2} + b x + c = 0$
$b x^{2} + c x + a = 0$ ,then
$\frac{α^{2}}{a b - c^{2}} = \frac{α}{c b - a^{2}} = \frac{1}{a c - b^{2}}$