Consider three points $$P = ( - \sin (\beta - \alpha ), - cos\beta ),Q = (cos(\beta - \alpha ),\sin \beta )$$ and $$R = (\cos (\beta - \alpha + \theta ),\sin (\beta - \theta ))$$ where $$0 < \alpha ,\beta ,\theta < {\pi \over 4}$$. Then :

Consider the lines given by:

$${L_1}:x + 3y - 5 = 0$$

$${L_2}:3x - ky - 1 = 0$$

$${L_3}:5x + 2y - 12 = 0$$

Match the Statement/Expressions in Column I with the Statements/Expressions in Column II.

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 :

Statement-1: The ratio $$PR$$ : $$RQ$$ equals $$2\sqrt 2 :\sqrt 5 $$. because
Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.