Question

If α ≠ β but α2 = 5α - 3 and β2 = 5β - 3, then the equation having α/β and β/α as its roots is

3x2 - 19x - 3 = 0
3x2 - 19x + 3 = 0
x2 - 5x + 3 = 0
3x2 + 19x - 3 = 0

Answer 1

Answer:

The correct option is B.

Step-by-step explanation:

The equation having α and β as its root will be:

x2 - (α+β)x + αβ = 0 and since α2 = 5α-3 and β2 = 5β-3 shows that α and β are roots of equation

x2 - 5x + 3 = 0 this implies α+β = 5 and αβ = 3

You can use these relations to calculate the equation having α/β and β/α as its root.  

(α/β)*(β/α) = 1 and  β/α + α/β = 19/3

Therefore, the equation is 3x2 - 19x + 3 = 0

Answer 2

Solution :-

We have

• α ≠ β

• α²= 5α - 3 → α² - 5α - 3 = 0

• β² = 5β - 3 → β² - 5β - 3 = 0

Replace x in place of α or β

→ x² - 5x - 3 = 0

So ,α and β are the roots of the equation.

→ α + β = 5

→ α β = 3

Now :-

equation which have α /β and β/α as roots

$\textsf{Sum of roots } = \dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}$

$= \dfrac{\alpha^2 + \beta^2}{\alpha\beta}$

$= \dfrac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta}$

$= \dfrac{(5)^2 - 2(3)}{3}$

$= \dfrac{25 - 6 }{3}$

$= \dfrac{19}{3}$

Product of roots = 1

So equation

→ x² - Sx + P = 0

$\rightarrow x^2 - \dfrac{19}{3}x + 1 = 0$

$\rightarrow 3x^2 - 19x + 3 = 0$

Option (B)