A line has intercepts a and b on the coordinate axes. When the axes are rotated through an angle $\alpha$, in the anti clockwise direction keeping the origin fixed, the line makes equal intercepts on the coordinate axes, then $\tan \alpha =$

Let the equation of the line in the original
coordinate system be $\frac{x}{a}+\frac{y}{b}=1$. If (X,Y)
 denote the coordinates
of any point P (x, y) in the new coordinate system
obtained by rotation of the axes through an angle a, then

$x=X\cos \alpha -Y\sin \alpha ,y=X\sin \alpha +Y\cos \alpha$

So that the equation of the line with reference to  new system of coordinates is

$\frac{X\cos \alpha -Y\sin \alpha }{a}+\frac{X\sin \alpha +Y\cos \alpha }{b}=1$

or $X\left(\frac{\cos \alpha }{a}+\frac{\sin \alpha }{b}\right)+Y\left(\frac{\cos \alpha }{b}-\frac{\sin \alpha }{a}\right)=1$

Since it makes equal intercepts on the coordinates axes.

$b\cos \alpha +a\sin \alpha =a\cos \alpha -b\sin \alpha$

$\Rightarrow (a-b)\cos \alpha =(a+b)\sin \alpha$

$\Rightarrow \tan \alpha =\frac{a-b}{a+b}$