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 $\tan \alpha =$

detailed solution

Correct option is B

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