Question

if $(a,a^2)$falls inside the angle made by the lines $y$ $=\frac{x}{2},x>0$ and $y=3x,x>0,$ then $\mathrm{a}$ belongs to

• A. $(3,\infty )$
• B. $(\frac{1}{2},3)$
• C. $(-3,\frac{-1}{2})$
• D. $(0,\frac{1}{2})$

Answer 1

The correct option is B

$(\frac{1}{2},3)$

Explanation for the correct option:

Finding the value:

The point $\left(a,a^2\right)$ lies inside the angle between two lines

$y$ $=\frac{x}{2},x>0$ and $y=3x,x>0,$

$\Rightarrow a^2–a/2>0$ and $a^2–3a>0$

$\Rightarrow a(a–\frac{1}{2})>0$ and $a(a–3)<0$

$\Rightarrow a>0$and $a>\frac{1}{2}$or $a<0$and $a<\frac{1}{2}$and $a<0,a>3$ or $a>0,a<3$

$\Rightarrow a>\frac{1}{2}$or $a<0$ and$0<a<3$

$\Rightarrow$$a\in (-\infty ,0)\cup (\frac{1}{2},\infty )$and $a\in (0,3)$

$\Rightarrow a\in (\frac{1}{2},3)$

Hence, option (B) is the correct answer.