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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2017 · Mathematics · Quadratic Equation

The sum of the roots of the equation ax^2+x+c=0 (where a and c are...

The sum of the roots of the equation ax2+x+c=0\mathrm{ax}^2+\mathrm{x}+\mathrm{c}=0 (where a and c are non-zero) is equal to the sum of the reciprocals of their squares. Then a, ca2,c2\mathrm{ca}^2, \mathrm{c}^2 are in

Options

  1. AAP
  2. BGP
  3. CHP
  4. DNone of the above

Correct answer

A. AP

Step-by-step solution

Let α,β\alpha, \beta be roots of ax2+x+c=0a x^2+x+c=0. Given, α+β=1α2+1β2=α2+β2α2β2=(α+β)2−2αβ(αβ)2\alpha+\beta=\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{\alpha^2+\beta^2}{\alpha^2 \beta^2}=\frac{(\alpha+\beta)^2-2 \alpha \beta}{(\alpha \beta)^2}⇒−1a=(−1a)2−2(ca)(ca)2⇒−1a=1a2−2cac2a2⇒−1a=1−2acc2⇒−c2=a−2a2c⇒2a2c=a+c2\begin{aligned} & \Rightarrow \frac{-1}{\mathrm{a}}=\frac{\left(\frac{-1}{\mathrm{a}}\right)^2-2\left(\frac{\mathrm{c}}{\mathrm{a}}\right)}{\left(\frac{\mathrm{c}}{\mathrm{a}}\right)^2} \\ & \Rightarrow \frac{-1}{\mathrm{a}}=\frac{\frac{1}{\mathrm{a}^2}-\frac{2 \mathrm{c}}{\mathrm{a}}}{\frac{\mathrm{c}^2}{\mathrm{a}^2}} \\ & \Rightarrow \frac{-1}{\mathrm{a}}=\frac{1-2 \mathrm{ac}}{\mathrm{c}^2} \Rightarrow-c^2=a-2 a^2 c \Rightarrow 2 a^2 c=a+c^2 \end{aligned}$$
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