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

The coordinates of the points $O$,$A$,and $B$ are $(0,0)$,$(0,4)$,and $(6,0)$ respectively. If a point $P$ moves such that the area of $\Delta POA$ is always twice the area of $\Delta POB$,then the equation to both parts of the locus of $P$ is

$A$ variable line passes through a fixed point $(a, b)$ and meets the coordinate axes at $A$ and $B$. The locus of the point of intersection of lines passing through $A$ and $B$ and parallel to the coordinate axes is:

Find the equation of a line passing through the point $(4, 3)$,which cuts a triangle of minimum area from the first quadrant.

If a point $(x, y) = (\tan \theta + \sin \theta, \tan \theta - \sin \theta)$,then the locus of $(x, y)$ is

Given $\frac{x}{a} + \frac{y}{b} = 1$ and $ax + by = 1$ are two variable lines,where $a$ and $b$ are parameters connected by the relation $a^2 + b^2 = ab$. The locus of the point of intersection has the equation: