If the equation to the locus of points equidistant from the points $(-2, 3)$ and $(6, -5)$ is $a x + b y + c = 0$,where $a > 0$,then the ascending order of $a, b, c$ is

The Cartesian form of the polar equation $\theta = \tan^{-1} 2$ is

Consider the triangles with vertices $A(2,1)$,$B(0,0)$,and $C(t,4)$,where $t \in [0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively,then $6\alpha + 21\beta$ is equal to $.........$.

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:

$A$ variable line $L$ passing through the origin cuts two parallel lines $x-y+10=0$ and $x-y+20=0$ at two points $A$ and $B$ respectively. If $P$ is a point on line $L$ such that $OA, OP, OB$ are in harmonic progression,then the locus of $P$ is

$A$ frog is presently located at the origin $(0,0)$ in the $XY$-plane. It always jumps from a point with integer coordinates to a point with integer coordinates,moving a distance of $5$ units in each jump. What is the minimum number of jumps required for the frog to go from $(0,0)$ to $(0,1)$?