Let the solution curve $$y=f(x)$$ of the differential equation $$ \frac{d y}{d x}+\frac{x y}{x^{2}-1}=\frac{x^{4}+2 x}{\sqrt{1-x^{2}}}$$, $$x\in(-1,1)$$ pass through the origin. Then $$\int\limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) d x $$ is equal to

The acute angle between the pair of tangents drawn to the ellipse $$2 x^{2}+3 y^{2}=5$$ from the point $$(1,3)$$ is :

The equation of a common tangent to the parabolas $$y=x^{2}$$ and $$y=-(x-2)^{2}$$ is

Let the abscissae of the two points $$P$$ and $$Q$$ on a circle be the roots of $$x^{2}-4 x-6=0$$ and the ordinates of $$\mathrm{P}$$ and $$\mathrm{Q}$$ be the roots of $$y^{2}+2 y-7=0$$. If $$\mathrm{PQ}$$ is a diameter of the circle $$x^{2}+y^{2}+2 a x+2 b y+c=0$$, then the value of $$(a+b-c)$$ is _____________.