question_answer

If \[(a,{{a}^{2}})\]falls inside the angle made by the lines y=\[\frac{x}{2}\], x>0, and \[y=3x,\text{ }x>0,\]then a belongs to

A) \[\left( 0,\frac{1}{2} \right)\]      

B) \[\left( 3,\,\infty  \right)\]

C) \[\left( \frac{1}{2},3 \right)\]      

D) \[\left( -3,-\frac{1}{2} \right)\]

Correct Answer: C

Solution :

[c] Given lines \[{{L}_{1}}:x-2y=0\] \[{{L}_{2}}:3x-y=0\] Points A and P lie on opposite Side of line \[x-2y=0.\] \[{{L}_{1}}(1,0)=1-2(0)>0\] \[\therefore a-2{{a}^{2}}<0\] Or \[a(2a-1)>0\] Or \[a>1/2\]                                           ...(i) Points A and P lie on same side of line \[3x-y=0\] \[{{L}_{2}}(1,0)=3-0>0\] \[\therefore 3a-{{a}^{2}}>0\] Or \[a(a-3)<0\] Or \[0<a<3\]                                         ...(ii) From (i) and (ii), \[1/2<a<3\]