# An object is placed to the left of a lens, and a real image is formed to the right of the lens. The image is inverted relative to the object and is one-half the size of the object. The distance between the object and the image is 65 cm. (a) How far from the lens is the object? (b) What is the focal length of the lens?

## To solve this problem, we can use the lens formula, which is:

1/f = 1/v - 1/u

where:
f is the focal length of the lens,
v is the image distance,
u is the object distance.

(a) To find the object distance (u), we are given that the image distance (v) is 65 cm. We can substitute these values into the lens formula and solve for u.

1/f = 1/65 - 1/u

Since the image is formed to the right of the lens, which means the image distance (v) is positive, we can substitute v = 65 cm into the lens formula:

1/f = 1/65 - 1/u

Now, let's solve for u:

1/u = 1/65 - 1/f

Since the image is half the size of the object, it means the magnification (M) is -1/2. The magnification is given by:

M = -v/u

Substituting the given values:

-1/2 = -(65/u)

Simplifying, we get:

1/2 = 65/u

Cross-multiplying, we have:

u = 2 * 65
u = 130 cm

Therefore, the object is located 130 cm to the left of the lens.

(b) Now that we have the object distance (u), we can use the lens formula to find the focal length (f). Substituting u = 130 cm and v = 65 cm into the lens formula:

1/f = 1/65 - 1/130

To add the fractions, we need a common denominator:

1/f = 2/130 - 1/130
1/f = 1/130

Therefore, the focal length of the lens is 130 cm.