(4) A converging (lens $$ \mathrm{f}=+12 \mathrm{~cm} $$ ) in contact with a diverging lens gives a combined focal length of +36 cm . Calculate the focal length of the diverging lens.

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**Step 1: Write the formula for the combination of thin lenses in contact**

For two lenses in contact, the combined focal length $$ f_c $$ is given by:  
$$
\frac{1}{f_c} = \frac{1}{f_1} + \frac{1}{f_2}
$$

**Step 2: Substitute the given values**

Given $$ f_1 = +12 $$ cm and $$ f_c = +36 $$ cm, substitute into the formula:  
$$
\frac{1}{36} = \frac{1}{12} + \frac{1}{f_2}
$$

**Step 3: Solve for $$ \frac{1}{f_2} $$**

Subtract $$ \frac{1}{12} $$ from both sides:  
$$
\frac{1}{f_2} = \frac{1}{36} - \frac{1}{12}
$$

Find a common denominator for the right side. Note that $$ \frac{1}{12} = \frac{3}{36} $$, so:  
$$
\frac{1}{f_2} = \frac{1}{36} - \frac{3}{36} = -\frac{2}{36}
$$

Simplify the fraction:  
$$
\frac{1}{f_2} = -\frac{1}{18}
$$

**Step 4: Find the focal length $$ f_2 $$**

Taking the reciprocal:  
$$
f_2 = -18 \text{ cm}
$$

**Final Answer**

The focal length of the diverging lens is $$ -18 $$ cm.
The focal length of the diverging lens is -18 cm.

(4) A converging (lens ) in contact with a diverging lens gives a combined focal
length of +36 cm . Calculate the focal length of the diverging lens.

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(4) A converging (lens ) in contact with a diverging lens gives a combined focal length of +36 cm . Calculate the focal length of the diverging lens.