Question:
A concave lens of focal length 15 cm forms an image 10 cm from the lens. How far is the object placed from the lens? Draw the ray diagram.

Answer:
To solve this problem, we can use the lens formula:

1/f = 1/v - 1/u

Where:
f is the focal length of the lens,
v is the image distance from the lens, and
u is the object distance from the lens.

Given:
Focal length (f) = -15 cm (since the lens is concave)
Image distance (v) = 10 cm

Step 1: Find the object distance (u)
We can rearrange the lens formula to solve for u:

1/u = 1/f - 1/v

Substituting the given values:

1/u = 1/(-15) - 1/10

Simplifying:

1/u = -2/30 - 3/30
= -5/30
= -1/6

Taking the reciprocal of both sides:

u = -6 cm

Step 2: Draw the ray diagram
To visualize the problem, we can draw a ray diagram. Follow these steps:

1. Draw a horizontal line representing the principal axis of the lens.
2. Mark the focal point (F) on the principal axis, which is 15 cm to the left of the lens.
3. Place an arrow on the principal axis to represent the object. The object is located 6 cm to the left of the lens.
4. Draw a ray from the tip of the object arrow parallel to the principal axis. After passing through the lens, this ray will appear to come from the focal point on the other side of the lens.
5. Draw a ray from the tip of the object arrow towards the center of the lens. This ray will pass straight through the lens without changing direction.
6. Draw an extension of the ray that passes through the center of the lens. This extension will intersect with the other ray at a point, which represents the location of the image.

Conclusion:
The object is placed 6 cm to the left of the concave lens.