$ABC$ is a variable triangle such that $A$ is $(1, 2)$,and $B$ and $C$ lie on the line $y = x + \lambda$ (where $\lambda$ is a variable). The locus of the orthocenter of triangle $ABC$ is:

$A (a, 0)$ and $B (-a, 0)$ are two fixed points of $\Delta ABC$. If the vertex $C$ moves such that $\cot A + \cot B = \lambda$,where $\lambda$ is a constant,then what is the locus of point $C$?

The locus of the point $P$ which is equidistant from $3x + 4y + 5 = 0$ and $9x + 12y + 7 = 0$ is:

$A$ line cuts the $x$-axis at $A(7, 0)$ and the $y$-axis at $B(0, -5)$. $A$ variable line $PQ$ is drawn perpendicular to $AB$ cutting the $x$-axis at $P(a, 0)$ and the $y$-axis at $Q(0, b)$. If $AQ$ and $BP$ intersect at $R(h, k)$, the locus of $R$ is

Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y$-axis at $C$. The locus of the mid-point $P$ of $MC$ is:

$A(2,3)$ 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