Question

If one of the equation ax² + bx + c = 0 is three times times the other, then b² : ac =

Options

• 3 : 1

• 3 : 16

• 16 : 3

• 16 : 1

Answer 1

Let `alpha and beta`be the roots of quadratic equation`ax^2 + bx + c = 0` in such a way that `alpha = 3beta`

Here, a = a, b = b and , c = c

Then,

according to question sum of the roots

`alpha + beta = (-b)/a`

`3 beta + beta = (-b)/a`

        `4beta = (-b)/a`

`alpha = (-b)/(4a)`….. (1)

And the product of the roots

`alpha . beta = c/a`

`3beta xx beta = c / a`

`3beta^2 = c/a`

`beta^2 = c /3a`….. (2)

Putting the value of `beta = (-b)/(4a)` in equation (2)

`(-b)/(4a)^2 = c /(3a)`

 `(b^2)/(16a^2) = c/3a`

       `b^2 = c/(3a) xx 16a^2`

`b^2 = (16ac)/3`

`b^2 / (ac) = 16/3`

`b^2 : ac = 16 :3`