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 a belongs to

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

Answer 1

The correct option is C $(\frac{1}{2},3)$


Given lines
$L_1:x-2y=0L_2:3x-y=0$
Points A and P lie on opposite side of line $x-2y=0L_1(1,0)=1-2\left(0\right)>0\therefore a-2a^2<0$
or $a(2a-1)>0$
or $a>\frac{1}{2}...\left(i\right)$
Points A and P lie on same side of line $3x-y=0L_2(1,0)=3-0>0\therefore 3a-a^2>0or(a-3)<0or0<a<3...\left(ii\right)$
From $\left(i\right)and\left(ii\right)\frac{1}{2}<a<3$