Let the line xcosα+ysinα=p touches the curve at point P(x1,y1)
The equation of tangent at P(x1,y1)
xx1a2−yy1b2=1 ...(1)
Here both the equation represent same line
Now the equation xcosα+ysinα=p
or xαcosα+ysinαp=1....(2)
Now coefficient of line (1) and (2) are same
∴x1a2=cosαp and −y1b2=sinαp
or, x1=a2cosαp and y1=−b2sinαp
∵ These two points lies on the equation of straight line
∴a2cos2αp−b2sin2αp=p
or, a2cos2α−b2sin2α=p2 hence prove