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Question : 32 of 39

Marks: +1, -0

If

\alpha

B

are the roots of the equation

3 x^{2}-5 x+3=0

then the value of

\;\frac{\alpha}{\beta}+\;\frac{\beta}{\alpha}

is

$\frac{7}{9}$
$-\frac{7}{9}$
$\frac{8}{9}$
$-\frac{8}{9}$

Solution:

Concept:

Use the sum and product of roots to simplify the expression

\frac{\alpha}{\beta} + \frac{\beta}{\alpha}

.

Explanation:

For

3x^2 - 5x + 3 = 0

, we have

a = 3

,

b = -5

,

c = 3

.

Sum of roots:

\alpha + \beta = -\frac{b}{a} = -\frac{(-5)}{3} = \frac{5}{3}

.

Product of roots:

\alpha\beta = \frac{c}{a} = \frac{3}{3} = 1

.

Now,

\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta}

.

\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(\frac{5}{3}\right)^2 - 2(1) = \frac{25}{9} - 2 = \frac{25}{9} - \frac{18}{9} = \frac{7}{9}

.

Thus,

\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{7}{9} \div 1 = \frac{7}{9}

.

Answer:

\frac{7}{9}

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