If $(\alpha, \beta)$ is the orthocentre of the triangle $ABC$ with vertices $A(3, -7)$,$B(-1, 2)$,and $C(4, 5)$,then $9\alpha - 6\beta + 60$ is equal to:

If the orthocentre of the triangle formed by the lines $2x+3y-1=0$,$x+2y-1=0$,and $ax+by-1=0$ is the centroid of another triangle,whose circumcentre and orthocentre are $(3,4)$ and $(-6,-8)$ respectively,then the value of $|a-b|$ is..........

The orthocenter of an equilateral triangle is $(3, -2)$. If one of its sides lies on the $x$-axis,find the vertex of the triangle that does not lie on the $x$-axis.

The coordinates of the orthocentre of the triangle whose sides are given by the lines $x = 3$,$y = 4$,and $3x + 4y = 6$ are:

The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points $(a^2 + 1, a^2 + 1)$ and $(2a, -2a)$,where $a \ne 0$. Then for any $a$,the orthocentre of this triangle lies on the line:

The incentre of the triangle formed by the straight lines $y=\sqrt{3}x$,$y=-\sqrt{3}x$ and $y=3$ is