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

If the lines represented by $$a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$$ intersect on the $$X$$-axis, which of the following is in general incorrect?

For $$\alpha \in\left[0, \frac{\pi}{2}\right]$$, the angle between the lines represented by $$[x \cos \theta-y] [(\cos \theta+\tan \alpha) x-(1-\cos \theta \tan \alpha) y]=0$$ is

When the axes are rotated through an angle 45$$^\circ$$, the new coordinates of a point P are (1, $$-$$1). The coordinates of P in the original system are