If the point $(a, a^2)$ lies inside the angle formed by the lines $y = \frac{x}{2}$ $(x > 0)$ and $y = 3x$ $(x > 0)$,then $a$ belongs to:

Let $A$ be the point $(0,4)$ in the $xy$-plane and let $B$ be the point $(2t, 0)$. Let $L$ be the midpoint of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $M$. Let $N$ be the midpoint of $LM$. Then the locus of $N$ is

Let $A=(4,0)$ and $B=(0,12)$ be two points in the plane. The locus of a point $C(x, y)$ such that the area of $\triangle ABC$ is $18$ sq units is

The locus of the centroid of the triangle whose vertices are $(a \cos t, a \sin t)$,$(b \sin t, -b \cos t)$,and $(1, 0)$,where $t$ is a parameter,is:

Two points $A$ and $B$ with coordinates $(1, 1)$ and $(-2, 3)$ are given respectively. Then,the locus of a point $P$ such that the area of $\triangle PAB$ is $9 \text{ sq. units}$ is given by $......$

$A$ line passing through the point $(1, 2)$ meets the axes at $P$ and $Q$ such that it forms a triangle $OPQ$,where $O$ is the origin. If the area of triangle $OPQ$ is minimum,then the slope of the line $PQ$ is: