Principal points, object for infinity image

Given: f₁ = f₂ = 8 cm, d = 4 cm.

Equivalent focal length

1/F = 1/8 + 1/8 − 4/(8×8) = 1/4 − 1/16 = 3/16
F   = 16/3 = 5.33 cm

Principal points

α = F d / f₂ = 5.33 × 4 / 8 = 2.67 cm  (H₁: 2.67 cm right of L1)
β = F d / f₁ = 5.33 × 4 / 8 = 2.67 cm  (H₂: 2.67 cm left of L2)

H₂ is 4 − 2.67 = 1.33 cm to the right of L1, so H₂ is nearer to L1 than H₁ is.

 L1        H2  H1          L2
 |---------.---.-----------|
 0        1.33  2.67       4.0   (cm from L1)

Object for an image at infinity

An image at infinity is formed when the object is at the first focal point F₁ of the system.

F₁ from L1 = α − F = 2.67 − 5.33 = −2.67 cm

So the object must be 2.67 cm in front of the first lens.

Check: for L1, u = −2.67 cm: 1/v₁ = 1/8 − 1/2.67 = −1/4, v₁ = −4 cm. This image is 4 + 4 = 8 cm to the left of L2, which is exactly f₂, so the final rays leave L2 parallel (image at infinity).

Answer: F = 5.33 cm; H₁ is 2.67 cm to the right of L1; H₂ is 2.67 cm to the left of L2; the object must be 2.67 cm in front of L1.