$A(2,3)$ and $B(3,-5)$ are two vertices of $\triangle ABC$. If the centroid of the $\triangle ABC$ moves on the line $2x+y-2=0$,then the locus of $C$ is

If $2a + b + 3c = 0$,then the line $ax + by + c = 0$ passes through the fixed point that is

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

$A$ point moves in the $XY$-plane such that the sum of its distances from two mutually perpendicular lines is always equal to $3$. The area enclosed by the locus of that point is (in sq. units)

$A(-4,0)$ and $B(4,0)$ are two fixed points. $C$ and $D$ are two points on the $Y$-axis such that $CD=4$ and $C$ is a point below $D$. Then the locus of the point of intersection of the lines $AC$ and $BD$ is

$A$ straight line $L$ cuts both the lines $5x - y - 4 = 0$ and $3x + 4y - 4 = 0$. The segment of $L$ between the two lines is bisected at the point $(1, 5)$. The equation of $L$ is