A line through $$A (-5, -4)$$ meets the line $$x + 3y + 2 = 0,$$ $$2x + y + 4 = 0$$ and $$x - y - 5 = 0$$ at the points $$B, C$$ and $$D$$ respectively. If $${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)^2},$$ find the equation of the line.

Determine all values of $$\alpha $$ for which the point $$\left( {\alpha ,\,{\alpha ^2}} \right)$$ lies insides the triangle formed by the lines $$$\matrix{ {2x + 3y - 1 = 0} \cr {x + 2y - 3 = 0} \cr {5x - 6y - 1 = 0} \cr } $$$

Show that all chords of the curve $$3{x^2} - {y^2} - 2x + 4y = 0,$$ which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

Find the equation of the line passing through the point $$(2, 3)$$ and making intercept of length 2 units between the lines $$y + 2x = 3$$ and $$y + 2x = 5$$.