Let $P$ be a moving point such that the sum of its perpendicular distances from the lines $2x + y - 3 = 0$ and $x - 2y + 1 = 0$ is always $2 \, units$. Then,the area bounded by the locus of point $P$ is:

Let the algebraic sum of the perpendicular distances from the points $A(2, 0)$,$B(0, 2)$,and $C(1, 1)$ to a variable line be zero. Then,all such lines:

The locus of the midpoint of the portion of the line $x \cos \alpha + y \sin \alpha = p$ intercepted by the coordinate axes,where $p$ is a constant,is

The locus of a point which moves such that the area of the triangle formed by it with the vertices $(1, 2)$ and $(-2, 5)$ is $8$ sq. units is/are

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

Let $A(-1,1)$ and $B(2,3)$ be two points and $P(x,y)$ be a variable point above the line $AB$ such that the area of $\triangle PAB$ is $10$. If the locus of $P$ is $ax+by=15$,then $5a+2b$ is: