The equation of the locus of a point which is equidistant from the points $(2, 3)$ and $(4, 5)$ is

If the equation of the locus of a point equidistant from $(a_1, b_1)$ and $(a_2, b_2)$ is $(a_1 - a_2)x + (b_1 - b_2)y + c = 0$,find the value of $c$.

If the sum of distances from a point $P$ to two mutually perpendicular straight lines is $1$ unit,then the locus of $P$ 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

If $A$ is $(2, 5)$,$B$ is $(4, -11)$ and $C$ lies on $9x + 7y + 4 = 0$,then the locus of the centroid of the $\Delta ABC$ is a straight line parallel to the straight line:

If $A(1,0), B(0,-2), C(2,-1)$ are three fixed points,then the equation of the locus of a point $P(x,y)$ such that the area of $\triangle PAB$ is equal to the area of $\triangle PAC$ is: