A convex lens is used to throw on a screen `10 m` from the lens, a magnified image of an object. If the magnification is to be `19`, find the focal length of the lens.

Explanation

To find the focal length of a convex lens given the image distance and magnification, we can follow these steps:

### Step 1: Understand the given values
- The distance from the lens to the screen (image distance, \(v\)) is given as \(10 \, m\).
- The magnification (\(m\)) is given as \(19\). Since the image is real and inverted, we will consider the magnification as negative, so \(m = -19\).

### Step 2: Use the magnification formula
The magnification (\(m\)) for a lens is given by the formula:
\[ m = -\frac{v}{u} \]
where \(v\) is the image distance and \(u\) is the object distance. Rearranging this gives:
\[ u = -\frac{v}{m} \]

### Step 3: Substitute the known values
Substituting \(v = 10 \, m\) and \(m = -19\):
\[ u = -\frac{10}{-19} = \frac{10}{19} \approx 0.526 \, m \]

### Step 4: Use the lens formula
The lens formula relates the focal length (\(f\)), object distance (\(u\)), and image distance (\(v\)):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

### Step 5: Substitute \(u\) and \(v\) into the lens formula
Substituting \(v = 10 \, m\) and \(u = \frac{10}{19} \, m\):
\[ \frac{1}{f} = \frac{1}{10} + \frac{19}{10} \]
Calculating the right side:
\[ \frac{1}{f} = \frac{1 + 19}{10} = \frac{20}{10} = 2 \]

### Step 6: Solve for \(f\)
Taking the reciprocal gives:
\[ f = \frac{1}{2} = 0.5 \, m \]

### Step 7: Convert to centimeters
Since \(1 \, m = 100 \, cm\):
\[ f = 0.5 \, m = 50 \, cm \]

### Final Answer
The focal length of the lens is \(50 \, cm\).

---