In $\triangle ABC$,if $A$ is $(1,2)$,and $B$ and $C$ lie on the line $y=x+\alpha$ (where $\alpha$ is a variable),then the locus of the orthocenter of the triangle is:

$AB$ is a line segment moving between the axes such that '$A$' lies on $X$-axis and '$B$' lies on $Y$-axis. If $P$ is a point on $AB$ such that $PA=b$ and $PB=a$,then the equation of the locus of $P$ is

The locus of the centroid of a triangle whose vertices are $(1, 0)$,$(a \cos t, a \sin t)$,and $(b \sin t, -b \cos t)$ is $9x^2 + 9y^2 - 6x = k$. Then,the value of $k$ is equal to

$ABC$ is an isosceles triangle. If the coordinates of the base are $B(1, 3)$ and $C(-2, 7)$,the coordinates of vertex $A$ can be

If $A(2,-3)$ and $B(-2,1)$ are two vertices of a triangle and the third vertex moves on the line $2x + 3y = 9$,then the locus of the centroid of the triangle is

If a variable line drawn through the intersection of the lines $\frac{x}{3} + \frac{y}{4} = 1$ and $\frac{x}{4} + \frac{y}{3} = 1$ meets the coordinate axes at $A$ and $B$ $(A \neq B)$,then the locus of the midpoint of $AB$ is