Find the distance of the image from object $$O$$, formed by the combination of lenses in the figure:

We need to find the total distance between the object $$O$$ and the final image produced by the combination of three thin lenses.

1. Analyze the First Lens (Convex Lens)

The object $$O$$ is placed to the left of the first convex lens ($$f_1 = +10\text{ cm}$$) at a distance of $$30\text{ cm}$$.

• Object distance, $$u_1 = -30\text{ cm}$$
• Focal length, $$f_1 = +10\text{ cm}$$

Applying the standard Lens Formula:

$$\frac{1}{v_1} - \frac{1}{u_1} = \frac{1}{f_1}$$

$$\frac{1}{v_1} - \frac{1}{-30} = \frac{1}{10} \implies \frac{1}{v_1} = \frac{1}{10} - \frac{1}{30}$$

$$\frac{1}{v_1} = \frac{3 - 1}{30} = \frac{2}{30} \implies v_1 = +15\text{ cm}$$

The first lens forms a real image $$15\text{ cm}$$ to its right.

2. Analyze the Second Lens (Concave Lens)

The second lens ($$f_2 = -10\text{ cm}$$) is placed $$5\text{ cm}$$ to the right of the first lens. The image from the first lens acts as a virtual object for this second lens.

• Object distance, $$u_2 = 15\text{ cm} - 5\text{ cm} = +10\text{ cm}$$
• Focal length, $$f_2 = -10\text{ cm}$$

Applying the Lens Formula for the second stage:

$$\frac{1}{v_2} - \frac{1}{u_2} = \frac{1}{f_2}$$

$$\frac{1}{v_2} - \frac{1}{10} = \frac{1}{-10} \implies \frac{1}{v_2} = -\frac{1}{10} + \frac{1}{10} = 0$$

$$v_2 = \infty$$

The light rays emerge completely parallel to the principal axis after passing through the second lens.

3. Analyze the Third Lens (Convex Lens)

The third lens ($$f_3 = +30\text{ cm}$$) is placed $$10\text{ cm}$$ to the right of the second lens. Since the incoming rays are traveling parallel to the principal axis ($$u_3 = \infty$$), they must naturally converge at its focus.

• Object distance, $$u_3 = \infty$$
• Focal length, $$f_3 = +30\text{ cm}$$

Applying the Lens Formula for the final stage:

$$\frac{1}{v_3} - \frac{1}{\infty} = \frac{1}{30} \implies v_3 = +30\text{ cm}$$

The final image is formed at a distance of $$30\text{ cm}$$ to the right of the third lens.

4. Calculate the Total Distance from Object $O$ to Final Image

The total geometric separation distance ($D$) across the entire setup is the sum of the initial object distance, all gaps between the lenses, and the final image distance:

$$D = (\text{Distance of } O \text{ to Lens 1}) + (\text{Gap 1}) + (\text{Gap 2}) + (\text{Distance of final image from Lens 3})$$

$$D = 30\text{ cm} + 5\text{ cm} + 10\text{ cm} + 30\text{ cm} = 75\text{ cm}$$

Final Answer: 75 cm