Question

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If alpha,beta are zeroes of the ax^2+bx+c,from a polynomial whose zeroes are 1/alpha and 1/beta.

_______Class 10th________

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Answer 1

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Answer 2

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Let p(x) = ax² + bx + c

Given

alpha and beta are the zeros of p(x)

Thus we know

$\alpha + \beta = \frac{ - b}{a}$

Similarly

$\alpha \beta = \frac{c}{a}$

Now let f(x) be the required Quadratic Polynomial

let S be the sum of the zeros and P be the product of zeros

Thus

S =

$\frac{1}{ \alpha } + \frac{1}{ \beta }$

$= \frac{ \alpha + \beta }{ \alpha \beta }$

Replacing values and cancelling a

S = - b/c

Similarly

P =
$\frac{1}{ \alpha \beta }$

Reciprocal of alpha x beta

Thus P = a/c

Thus we know

=> f(x) = k(x² - Sx + P)

$= k( {x}^{2} + \frac{b}{c}x + \frac{a}{c} )$

Taking 1/c as common

$= \frac{1}{c} (c {x}^{2} + bx + a)$

Thus the required Quadratic Polynomial is cx² + bx + a

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