Question

if one root of the equation (k-1)x²-2m x+3=0 is the reciprocal of the other then what is the value of k and m​

Answer 1

Concept Introduction: Quadratic equations are very special equations.

Given:

We have been Given:

$(k - 1) {x}^{2} - 2mx + 3 = 0$

One root of the equation is the reciprocal of the other.

To Find:

We have to Find: The Values of

$k \: and \: m$

Solution:

According to the problem, One root of the equation is the reciprocal of the other, so the let the roots be

$y \: and \: \frac{1}{y}$

therefore their products are

$1$

therefore:

$\frac{2m}{k - 1} = 1 \\ 2m = k - 1$

therefore putting the value of

$k - 1$

in the original equation, we get

$2m {x}^{2} - 2m + 3 = 0 \\ 2m( {x}^{2} - 1) = - 3 \\ m = - \frac{3 }{ {4x}^{2} - 2 }$

therefore,

$k = 1 - \frac{3}{ {x}^{2} - 1} \\ \frac{ {x}^{2} - 1 - 3}{ {x}^{2} - 1} = \frac{ {x }^{2} - 4}{ {x}^{2} - 1}$

therefore putting the values of k and m in original equation,

$\frac{ - 3 {x}^{2} }{ {x}^{2} - 1} + \frac{3x}{ {x}^{2} - 1} + 3 = 0 \\ \frac{3x - 3 {x}^{2} }{ {x}^{2} - 1} = - 3 \\ 3x - 3 {x}^{2} = 3 - 3 {x}^{2} \\ 3x = 3 \\ x = 1$

therefore putting the value of x in k and m we get,

$m = k = 0$

Final Answer: The Values if k and m are

$0$

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