Let a and b be non-zero real numbers. Then, the equation

$$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$ represents :

Let $$\mathrm{O(0,0), P(3,4), Q(6,0)}$$ be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

Lines $$\mathrm{L}_{1}: y-x=0$$ and $$\mathrm{L}_{2}: 2 x+y=0$$ intersect the line $$\mathrm{L}_{3}: y+2=0$$ at $$\mathrm{P}$$ and $$\mathrm{Q}$$, respectively. The bisector of the acute angle between $$L_{1}$$ and $$L_{2}$$ intersects $$L_{3}$$ at $$R$$.

STATEMENT - 1 : The ratio PR : RQ equals $$2 \sqrt{2}: \sqrt{5}$$.

STATEMENT - 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

Consider the following linear equations

$$ax + by + cz = 0$$

$$bx + cy + az = 0$$

$$cx + ay + bz = 0$$

Match the conditions/expressions in Column I with statements in Column II.