If one of the equation ax2 + bx + c = 0 is three times times

Solution:

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

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

Then,

according to question sum of the roots

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

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

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

$\beta=\frac{-b}{4 a} .$.....(1)

And the product of the roots

$\alpha \cdot \beta=\frac{c}{a}$

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

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

$\beta^{2}=\frac{c}{3 a} \ldots$(2)

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

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

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

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

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

$b^{2}: a c=16: 3$

Thus, the correct answer is $(c)$