Q173 / 270 · NDA 2012 (Phase 1) · Mathematics

For the next Two (2) questions that follow: The equation formed by multiplying each root of $a x^2+b x+c=0$ by 2 is $x^2+36 x+24=0$. The equation formed by multiplying each root of $a x^2+b x$ $+c=0$ by 2 is $x^2+36 x+24=0$. What is $b: c$ equal to?

  • Correct answer
    A
    $3: 1$
  • B
    $1: 2$
  • C
    $1: 3$
  • D
    $3: 2$

Solution

Let $\alpha, \beta$ be roots of equation $\begin{aligned} a x^2+b x+c & =0 \\ \therefore \quad \alpha+\beta & =-b / a \\ \alpha \beta & =c / a \end{aligned}$ So, $2 \alpha$ and $2 \beta$ we roots of equation $\begin{aligned} x^2+36 x+24 & =0 \\ 2 \alpha+2 \beta & =-36 \\ \alpha+\beta & =-18 \\ \Rightarrow \quad \frac{-b}{a} & =-18 \Rightarrow b=18 a \\ \Rightarrow \quad(2 \alpha)+(2 \beta) & =24 \\ \alpha \beta & =6 \\ \frac{c}{a} & =6 \Rightarrow c=6 a \\ \therefore b: c=3: & 1 . \end{aligned}$

Hence, the correct answer is option: 'A'.

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