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JEE Main 201226 MayOnline
Exam
JEE Main
Year
2012
Date
26 May
Mode
Online
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2012 · 26 May · Online · Mathematics · Quadratic Equation

If a, b, c ∈ R and 1 is a root of equation a x^2+b x +c=0 , then...

If a,b,c∈Ra, b, c \in \mathrm{R} and 1 is a root of equation ax2+bxa x^2+b x +c=0+c=0, then the curve y=4ax2+3bx+2c,a≠0y=4 a x^2+3 b x+2 c, a \neq 0 intersect xx-axis at

Options

  1. Atwo distinct points whose coordinates are always rational numbers
  2. Bno point
  3. Cexactly two distinct points
  4. Dexactly one point

Correct answer

D. exactly one point

Step-by-step solution

Given ax2+bx+c=0a x^2+b x+c=0⇒ax2=−bx−c Now, consider y=4ax2+3bx+2c=4[−bx−c]+3bx+2c=−4bx−4c+3bx+2c=−bx−2c\begin{aligned}& \Rightarrow a x²=-b x-c \\ & \text{ Now, consider }\\ & y=4 a x²+3 b x+2 c \\ & =4[-b x-c]+3 b x+2 c \\ & =-4 b x-4 c+3 b x+2 c \\ & =-b x-2 c \end{aligned}Since,thiscurveintersectsSince, this curve intersectsx−axis-axis\thereforeputputy=0,weget, we get−bx−2c=0⇒−bx=2c⇒x=−2cb\begin{aligned}& -b x-2 c=0 \Rightarrow-b x=2 c \\ & \Rightarrow x=\frac{-2 c}{b} \end{aligned}Thus,givencurveintersectsThus, given curve intersectsx$-axis at exactly one point.
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