The vertices of a triangle are
$$\left[ {a{t_1}{t_2},\,\,a\left( {{t_1} + {t_2}} \right)} \right],\,\,\left[ {a{t_2}{t_3},a\left( {{t_2} + {t_3}} \right)} \right],\,\,\left[ {a{t_3}{t_1},\,a\left( {{t_3} + {t_1}} \right)} \right]$$. Find the orthocentre of the triangle.

The coordinates of $$A, B, C$$ are $$(6, 3), (-3, 5), (4, -2)$$ respectively, and $$P$$ is any point $$(x, y)$$. Show that the ratio of the area of the triangles $$\Delta $$ $$PBC$$ and $$\Delta $$$$ABC$$ is $$\left| {{{x + y - 2} \over 7}} \right|$$

The end $$A, B$$ of a straight line segment of constant length $$c$$ slide upon the fixed rectangular axes $$OX, OY$$ respectively. If the rectangle $$OAPB$$ be completed, then show that the locus of the foot of the perpendicular drawn from $$P$$ to $$AB$$ is $${x^{{2 \over 3}}} + {y^{{2 \over 3}}} = {c^{{2 \over 3}}}$$

A straight line $$L$$ is perpendicular to the line $$5x - y = 1.$$ The area of the triangle formed by the line $$L$$ and the coordinate axes is $$5$$. Find the equation of the Line $$L$$.