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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2019 · Mathematics · Quadratic Equation

If the roots of the equation a(b-c) x^2+b(c-a) x+c(a-b)=0 are...

If the roots of the equation a(b−c)x2+b(c−a)x+c(a−b)=0a(b-c) x^2+b(c-a) x+c(a-b)=0 are equal, then which one of the following is correct?

Options

  1. Aa,ba, b and cc are in AP
  2. Ba,ba, b and cc are in GP
  3. Ca,ba, b and cc are in HP
  4. Da,ba, b and cc do not follow any regular pattern

Correct answer

C. a,ba, b and cc are in HP

Step-by-step solution

a(b−c)x2+b(c−a)x+c(a−b)=0a(b-c) x^2+b(c-a) x+c(a-b)=0 has coefficients in cyclic order ∴\therefore its one root will be 1. ∵\because both the roots are equal ∴\therefore other root =1=1∴ Product of roots =1⇒c(a−b)a(b−c)=1⇒ac−bc=ab−ac⇒2ac=ab+bc⇒2b=1a+1c⇒a,b,c are in HP. \begin{aligned} & \therefore \text{ Product of roots }=1 \\ & \Rightarrow \frac{c(a-b)}{a(b-c)}=1 \\ & \Rightarrow a c-b c=a b-a c \\ & \Rightarrow 2 a c=a b+b c \\ & \Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c} \\ & \Rightarrow a, b, c \text{ are in HP. }\end{aligned}$$
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