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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2013 · Mathematics · Quadratic Equation

If the roots of the equation 3 a x^2+2 b x+c=0 are in the ratio 2:...

If the roots of the equation 3ax2+2bx+c=03 a x^2+2 b x+c=0 are in the ratio 2:32: 3, then which one of the following is correct?

Options

  1. A8ac=25b8 a c=25 b
  2. B8ac=9b28 a c=9 b^2
  3. C8b2=9ac8 b^2=9 a c
  4. D8b2=25ac8 b^2=25 a c

Correct answer

D. 8b2=25ac8 b^2=25 a c

Step-by-step solution

Let roots of equation 3ax2+2bx+c=03 a x^2+2 b x+c=0 are 2α,3α2α+3α=−2b3a⇒α=−2b15a2α⋅3α=c3a⇒α2=c18a2 \alpha, 3 \alpha2 \alpha+3 \alpha=-\frac{2 b}{3 a} \quad \Rightarrow \quad \alpha=-\frac{2 b}{15 a}2 \alpha \cdot 3 \alpha=\frac{c}{3 a} \quad \Rightarrow \quad \alpha^2=\frac{c}{18 a}∴(−2b15a)2=c18a⇒4b2225a2=c18a8b2=25ac\begin{aligned} \therefore \quad\left(\frac{-2 b}{15 a}\right)^2 & =\frac{c}{18 a} \Rightarrow \frac{4 b^2}{225 a^2}=\frac{c}{18 a} \\ 8 b^2 & =25 a c\end{aligned}$$
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