If two lines represented by $$a x^2+2 h x y+b y^2=0$$ makes angles $$\alpha$$ and $$\beta$$ with positive direction of $$\mathrm{X}$$-axis, then $$\tan (\alpha+\beta)=$$

If $$\tan ^{-1}\left[\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right]=\alpha$$, then the value of $$\sin 2 \alpha$$ is

The general solution of the differential equation $$\frac{d y}{d x}=2^{y-x}$$ is

The vector equation of the line passing through $$\mathrm{P}(1,2,3)$$ and $$\mathrm{Q}(2,3,4)$$ is