$A$ 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 drawn through $A$ and $B$ parallel to the coordinate axes is:

$A$ straight line meets the $X$ and $Y$ axes at the points $A$ and $B$ respectively. If $AB = 6$ units,then the locus of the point $P$ which divides the line segment $AB$ such that $AP : PB = 2 : 1$ is

$A$ variable line passing through $(l, m)$ intersects the coordinate axes at the points $A$ and $B$. If the line drawn parallel to $Y$-axis through $A$ and parallel to $X$-axis through $B$ meet at $P$,then the locus of $P$ is

The point moves such that the area of the triangle formed by it with the points $(1, 5)$ and $(3, -7)$ is $21$ sq. units. The locus of the point is:

$A$ straight line $L$ with negative slope passes through the point $(1,1)$ and cuts the positive coordinate axes at the points $A$ and $B$. If $O$ is the origin,then the minimum value of $OA + OB$ as $L$ varies,is

Let $A(2,1)$ be a point and the equation of the straight line $L$ be $x-y=0$. Let $a$ and $b$ respectively represent the distances from a variable point $P(\alpha, \beta)$ to $A$ and to the line $L$. If $c$ is the distance of the point $A$ from the origin such that $a=bc$,then the locus of $P$ is