If the straight line xcosα + ysinα = p touches the curve then prove that
a2cos2α–b2sin2α = ρ2.
Given:
The straight line xcosα + ysinα = p touches the curve 1.
Suppose the straight line xcosα + ysinα = p touches the curve at (x1,y1).
But the equation of tangent to 1 at (x1,y1) is
1
Thus ,equation 1 and xcosα + ysinα = p represent the same line.
∴
⇒ x1 ,y1
Since the point (x1,y1) lies on the curve 1
⇒ 1
⇒ 1
⇒ 1
⇒ a2cos2α – b2sin2α = p2
Thus proved.