Which of the following pair of equations represent the same conic-

Explanation -

The equation of the given curve \(\frac{l}{r} = 1 + e\cos θ \) .....(i)

Now (r, θ) and (-r, θ + π) represent the same point.

So replace r  by -r and θ by θ + π  in the equation 1

We get -

⇒ \(\frac{l}{-r} = 1 + e\cos (θ+\pi) \)

We know that cos(θ + π) = - cosθ 

⇒ \(\frac{l}{-r} = 1 - e\cos (θ) \)

⇒ \(\frac{l}{r} = -1 + e\cos (θ) \) .....(2) 

Therefore equation (1) and (2) represents the same conic.

Hence \(\frac{l}{r} = 1 + e\cos θ \) &  \(\frac{l}{r} = -1 + e\cos (θ) \) represent the same conic.