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

Let the equation of the line in the originalcoordinate system be xa+yb=1. If (X,Y) denote the coordinatesof any point P (x, y) in the new coordinate systemobtained by rotation of the axes through an angle a, thenx=Xcos⁡α−Ysin⁡α,y=Xsin⁡α+Ycos⁡αSo that the equation of the line with reference to  new system of coordinates isXcos⁡α−Ysin⁡αa+Xsin⁡α+Ycos⁡αb=1or Xcos⁡αa+sin⁡αb+Ycos⁡αb−sin⁡αa=1Since it makes equal intercepts on the coordinates axes.bcos⁡α+asin⁡α=acos⁡α−bsin⁡α⇒(a−b)cos⁡α=(a+b)sin⁡α⇒ tan⁡α=a−ba+b