Optic axis of a thin equi-convex lens is the x-axis. The co-ordinates of a point object and its image are `(-40 cm,1 cm) and (50 cm,-2 cm),` respectively. Lens is located at

Options

A. `x=20 cm`
B. `x=-30 cm`
C. `x=-10 cm`
D. origin

Explanation

To find the location of the lens based on the given coordinates of the object and its image, we can follow these steps:

### Step 1: Identify the coordinates
The coordinates of the point object and its image are given as:
- Object (O): (-40 cm, 1 cm)
- Image (I): (50 cm, -2 cm)

### Step 2: Determine the heights
From the coordinates, we can identify the heights:
- Height of the object (H_object) = 1 cm
- Height of the image (H_image) = -2 cm (the negative sign indicates that the image is inverted)

### Step 3: Calculate the magnification (m)
Magnification (m) is given by the formula:
\[ m = \frac{H_{image}}{H_{object}} \]
Substituting the values:
\[ m = \frac{-2}{1} = -2 \]

### Step 4: Relate object distance (u) and image distance (v)
The magnification can also be expressed in terms of object distance (u) and image distance (v):
\[ m = \frac{-v}{u} \]
From the magnification calculated:
\[ -2 = \frac{-v}{u} \]
This leads to:
\[ v = 2u \]

### Step 5: Determine the distances
Let the distance of the object from the lens be \( d_1 \) and the distance of the image from the lens be \( d_2 \). According to the problem:
\[ d_1 + d_2 = 90 \text{ cm} \]
Since \( d_1 = -u \) and \( d_2 = v \), we can substitute:
\[ -u + 2u = 90 \]
This simplifies to:
\[ u = 90 \text{ cm} \]

### Step 6: Calculate the image distance (v)
Now substituting \( u \) back into the equation for \( v \):
\[ v = 2u = 2 \times 90 = 180 \text{ cm} \]

### Step 7: Determine the location of the lens
The object is located at \( -40 \text{ cm} \), and we have found \( u = 30 \text{ cm} \) (the distance from the lens to the object). Thus, the lens is located at:
\[ \text{Lens position} = -40 \text{ cm} + 30 \text{ cm} = -10 \text{ cm} \]

### Conclusion
Therefore, the lens is located at \( x = -10 \text{ cm} \).

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