A line passing through $$P(2,3)$$ and making an angle of $$30^{\circ}$$ with the positive direction of $$X$$-axis meets $$x^2-2 x y-y^2=0$$ at $$A$$ and $$B$$. Then the value of $$P A: P B$$ is

The least distance from origin to a point on the line $$y=x+3$$ which lies at a distance of 2 units from $$(0,3)$$ is

Starting from the point $$A(-3,4)$$, a moving object touches $$2 x+y-7=0$$ at $$B$$ and reaches the point $$C(0,1)$$. If the object travels along the shortest path, the distance between $$A$$ and $$B$$ is

Suppose a triangle is formed by $$x+y=10$$ and the coordinate axes. Then, the number of points $$(x, y)$$ where $$x$$ and $$y$$ are natural numbers, lying inside the triangle is