Question

If the points $(a,a^2)$ and $(1,2)$ lie in the same angular region between the lines $3x+4y-1=0$ and $2x+y-3=0$, then

• A. $a<-3$ or $a>1$
• B. $aϵ[-3,1]$
• C. $a<\frac{1}{2}$ or $a>-1$
• D. None of these

Answer 1

Answer: (A)

$a<-3$ or $a>1$

Given points will lie in the same angular region between the lines $3x+4y-1=0$ and $2x+y-3=0$, if they lie on the same side of the two lines.
$\therefore (3a+4a^2-1)(3+8-1)>0$ and $(2a+a^2-3)(2+2-3)>0$
$\Rightarrow 4a^2+3a-1>0$ and $a^2+2a-3>0$
$\Rightarrow (4a-1)(a+1)>0$ and $(a+3)(a-1)>0$
$\Rightarrow (a<-1ora>\frac{1}{4})$ and $(a<-3ora>1)$
$\Rightarrow a<-3ora>1$