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 ${\displaystyle \alpha \text{and}\beta }$be the roots of quadratic equation${\displaystyle a{x}^2+bx+c=0}$ in such a way that ${\displaystyle \alpha =3\beta }$

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

Then,

according to question sum of the roots

${\displaystyle \alpha +\beta =\frac{-b}{a}}$

${\displaystyle 3\beta +\beta =\frac{-b}{a}}$

        ${\displaystyle 4\beta =\frac{-b}{a}}$

${\displaystyle \alpha =\frac{-b}{4a}}$….. (1)

And the product of the roots

${\displaystyle \alpha .\beta =\frac{c}{a}}$

${\displaystyle 3\beta \times \beta =\frac{c}{a}}$

${\displaystyle 3{\beta }^2=\frac{c}{a}}$

${\displaystyle {\beta }^2=\frac{c}{3}a}$….. (2)

Putting the value of ${\displaystyle \beta =\frac{-b}{4a}}$ in equation (2)

${\displaystyle \frac{-b}{{\left(4a\right)}^2}=\frac{c}{3a}}$

 ${\displaystyle \frac{b^2}{16a^2}=\frac{c}{3}a}$

       ${\displaystyle b^2=\frac{c}{3a}\times 16a^2}$

${\displaystyle b^2=\frac{16ac}{3}}$

${\displaystyle \frac{b^2}{ac}=\frac{16}{3}}$

${\displaystyle b^2:ac=16:3}$