Let $P(2, 3)$,$Q(6, 0)$,and $R(\alpha, \beta)$ be three points in the $x-y$ plane such that $|PR + QR| + |PR - QR|$ is minimum. Then the value of $(\alpha - 2\beta)$ is:

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:

Show that the path of a moving point such that its distances from two lines $3x - 2y = 5$ and $3x + 2y = 5$ are equal is a straight line.

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:

Let $S$ be the set of points whose abscissae and ordinates are natural numbers. Let $P \in S$ be such that the sum of the distances of $P$ from $(8,0)$ and $(0,12)$ is minimum among all elements in $S$. Then, the number of such points $P$ in $S$ is

$A$ line cuts the $X$-axis at $A(5,0)$ and the $Y$-axis at $B(0,-3)$. $A$ variable line $PQ$ is drawn perpendicular to $AB$ cutting the $X$-axis at $P$ and the $Y$-axis at $Q$. If $AQ$ and $BP$ meet at $R$, then the locus of $R$ is