The locus of a point $P(x, y)$ which moves in such a way that the segment $OP$,where $O$ is the origin $(0, 0)$,has a slope of $\sqrt{3}$ is:

$A$ variable line through the point $P(-1, 2)$ cuts the coordinate axes at $A$ and $B$ respectively. If $Q$ is a point on $AB$ such that $PA, PQ, PB$ are in a harmonic progression,then the locus of $Q$ is

$A$ straight line passing through a fixed point $(2, 3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $OPRQ$ is completed,then the locus of $R$ is:

If a variable line is moving such that the intercepts made by it on the coordinate axes are reciprocal to each other,then the points $P(x, y)$ on such lines satisfy

For three consecutive odd integers $a, b$ and $c$,if the variable line $a x+b y+c=0$ always passes through the point $(\alpha, \beta)$,the value of $\alpha^2+\beta^2$ equals

If for a variable line $\frac{x}{a} + \frac{y}{b} = 1$,the condition $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$ ($c$ is a constant) is satisfied,then the locus of the foot of the perpendicular drawn from the origin to the line is: