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 (x_1,y₁).

But the equation of tangent to 1 at (x_1,y₁) is

1

Thus ,equation 1 and xcosα + ysinα = p represent the same line.

∴

⇒ x₁ ,y₁

Since the point (x_1,y₁) lies on the curve 1

⇒ 1

⇒ 1

⇒ 1

⇒ a²cos²α – b²sin²α = p²

Thus proved.