If $a, b, c$ are in harmonical progression,then the line $bcx + cay + ab = 0$ passes through a fixed point whose coordinates are:

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:

The locus of the mid-points of the perpendiculars drawn from points on the line $x=2y$ to the line $x=y$ is:

$A$ variable line passes through a fixed point $(a, b)$ and meets the coordinate axes at $A$ and $B$. The locus of the point of intersection of lines passing through $A$ and $B$ and parallel to the coordinate axes is:

$A$ point $P$ on a line is at a distance of $4$ units from the origin $(0,0)$. If the line makes an angle of $60^{\circ}$ with the negative direction of the $X$-axis,then the coordinates of $P$ are

Let $A(5, -3)$,$B(3, -2)$,and $C(-1, 5)$ be three points. If $P$ is a point satisfying the condition $PA^2 + 2PB^2 = 3PC^2$,then a point that lies on the locus of $P$ is