The orthocentre of the triangle whose vertices are $(0, 0)$,$(3, 0)$,and $(0, 4)$ is

If the vertices $P, Q, R$ of a triangle $PQR$ are rational points,which of the following points of the triangle $PQR$ is (are) always a rational point $(s)$? ($A$ rational point is a point both of whose coordinates are rational numbers.)

Let $k$ be an integer such that the triangle with vertices $(k, -3k)$,$(5, k)$,and $(-k, 2)$ has an area of $28$ sq. units. Then the orthocentre of this triangle is at the point:

Two vertices of a triangle $ABC$ are $A(3, -1)$ and $B(-2, 3)$,and its orthocentre is $P(1, 1)$. If the coordinates of the point $C$ are $(\alpha, \beta)$ and the centre of the circle circumscribing the triangle $PAB$ is $(h, k)$,then the value of $(\alpha + \beta) + 2(h + k)$ equals :

The orthocentre of the triangle with vertices $(-2, -6)$,$(-2, 4)$ and $(1, 3)$ is

In a triangle $ABC$,the coordinates of $A$ are $(1, 2)$ and the equations of the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$ respectively. Then the area of $\Delta ABC$ (in sq. units) is