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 $$y + 2x = 5$$.

A line cuts the $$x$$-axis at $$A (7, 0)$$ and the $$y$$-axis at $$B (0, -5)$$. A variable line $$PQ$$ is drawn perpendicular to $$AB$$ cutting the $$x$$axis in $$P$$ and they $$Y$$-axis in $$Q$$. If $$AQ$$ and $$BP$$ intersect at $$R$$, find the locus of R.