Question

Solve this question

Also check my other question and help me out​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

$\red{\rm :\longmapsto\:\alpha,\beta \: be \: the \: roots \: of \: required \: quadratic \: equation}$

Given that,

↝ Arithmetic mean (A.M.) between the roots is 8/5.

We know,

By Definition of Arithmetic mean

If a and b are two numbers, then A.M. between them is

$\rm :\longmapsto\:\boxed{ \tt{ \: A.M. \: = \: \frac{a + b}{2} \: }}$

So, using this, we have

$\rm :\longmapsto\:\dfrac{\alpha + \beta}{2} = \dfrac{8}{2}$

$\bf\implies \:\boxed{ \tt{ \: \alpha + \beta = \frac{16}{5} \: }} - - - - (1)$

Also, given that

↝ A.M. between the Reciprocal of the roots is 8/7.

So,

$\rm :\longmapsto\:\dfrac{1}{2} \bigg[\dfrac{1}{\alpha} + \dfrac{1}{\beta} \bigg] = \dfrac{8}{7}$

$\rm :\longmapsto\:\dfrac{\beta + \alpha}{\alpha \: \beta} = \dfrac{16}{7}$

can be rewritten as using equation (1)

$\rm :\longmapsto\:\dfrac{16}{5(\alpha \: \beta)} = \dfrac{16}{7}$

$\rm :\longmapsto\:\dfrac{1}{5(\alpha \: \beta)} = \dfrac{1}{7}$

$\rm \implies\:\boxed{ \tt{ \: \alpha \: \beta = \frac{7}{5} \: }} - - - - (2)$

Now, we know, Quadratic equation is given by

$\green{\rm :\longmapsto\:\boxed{ \tt{ \: {x}^{2} \: - \: (\alpha + \beta)x \: + \: \alpha \: \beta \: = \: 0 \: }}}$

So, on substituting the values from equation (1) and (2), we get

$\rm :\longmapsto\: {x}^{2} - \bigg[\dfrac{16}{5} \bigg]x + \bigg[\dfrac{7}{5} \bigg] = 0$

$\rm :\longmapsto\: {5x}^{2} - 16x + 7 = 0$

Hence, the required Quadratic equation is

$\red{\rm :\longmapsto\:\boxed{ \bf{ \: {5x}^{2} \: - \: 16x \: + \: 7 \: = \: 0 \: }}}$

So,

• Option (a) is correct.

More to know :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac