Question

Solve the above question
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Answer 1

Need to FinD :-

• The value of k .

$\red{\frak{Given}}\begin{cases}\textsf{ One root of the equation is reciprocal of other .}\\\\\sf The \ equation\ is \ 4x^2 - 2x + ( k - 4 ) = 0 \end{cases}$

Given that the root of a quadratic equation are reciprocal of each other. So , let us take one root of the equation be α , then the other root will be 1/α . Now the quadratic equation is ,

$\sf\longrightarrow 4x^2 - 2x + ( k -4) = 0$

On comparing it to the Standard form of the equation which is ,

$\sf\longrightarrow ax^2 + bx + c = 0$

We get ,

• a = 4 , b = (-2) and c = ( k - 4)

Now , we know that the product of roots of a quadratic equation is given by ,

$\sf\longrightarrow \purple{Product_{(of \ roots )}= \dfrac{c}{a}}$

Therefore ,

$\sf\longrightarrow Product_{(of \ roots )}= \dfrac{c}{a} \\\\\\\sf\longrightarrow \alpha \times \dfrac{1}{\alpha}= \dfrac{ k - 4 }{4}\\\\\\\sf\longrightarrow 1 = \dfrac{k-4}{4}\\\\\\\sf\longrightarrow 4 = k - 4 \\\\\\\sf\longrightarrow k = 4 + 4 \\\\\\\sf\longrightarrow \underline{\underline{\red{k = 8 }}}$

Option B is correct .