Q:

How to find the orthocentre of a triangle whose coordinates and area are given

I have a triangle in $ xy $ plane, whose vertices being
$$ A(k, –3k)$$
$$ B(5, k) $$
$$ C(–k, 2) $$
Where $k$ is an integer.
And area is $28$ sq. units.
How to find the coordinates of its orthocentre?
 What I've done
By using area formula, I got $k=2$ 

A:

Hint
Since the area is given as well as the coordinates, this can help you determine the value of $k$ from the following:
$$\frac{1}{2}\begin{vmatrix}k&-3k&1\\5&k&1\\-k&2&1\end{vmatrix}=28.$$
Once you have the $k$, it is a matter of determining the intersection of two altitudes of the triangle. 
For example, to get the altitude from $C$ onto $AB$, you want to find a line that passes through $C$ and is perpendicular to $AB$. Slope of $AB$ is $m_{AB}=\frac{4k}{5-k}$. So your line must of the form
$$y-2=-\frac{5-k}{4k}(x+k).$$
Similarly you can get the other altitude (say from $B$ onto $AC$) and then find the intersection of these two altitudes. By definition, the intersection is the orthocenter. 

