Question

If ρ₁, ρ₂ are the radii of curvature at the extremities of focal chord of the conic I/r = (1 + e cosθ), prove that when e = 1, ρ₁ ^–2/3 + ρ₂ ^–2/3 = l ^–2/3. 

Answer 1

Let r = 1/u                                                                                                         ......(1)

so that for l/r = (1 + e cosθ) we get

and 

                                      .....(5)

The general equation of the conic I/r = (1 + e cos θ) represents

 a parabola for e = 1

 an ellipse for e < 1

 a hyperbola for e > 1

thus for e =1 , p = 

                                                        .....(6)

Now if ρ at P is termed as ρ₁ and ρ at Q is termed as ρ₂, then

Add the two, ρ₁ ^–2/3 + ρ₂ ^–2/3 = l ^–2/3