Enlarged or Height of the object be = h.
Height of the image = -4h.

Given,

u = -8 cm,
v = ?,
R = ?.


Solution:-


Height of the image (-4h)/Height of the object (h) =
-v/u.

-4h/h = -v/u.

4u = v.

v = 4 × -8.

v = -32 cm.

We get the focal length as -32 cm due to the answer for "v" is -32 cm.

f = -32cm.

We should find the Radius of Curvature, so, use the formula R = 2f.

R = 2f.
R = 2×-32.
R = -64 cm.

Given information:
- The convex mirror produces a 4 times enlarged image.
- The object is placed 8 cm in front of the convex mirror.

Formula:
The magnification produced by a convex mirror is given by the formula:

magnification (m) = -v/u

Where:
- v is the image distance (distance of the image from the mirror)
- u is the object distance (distance of the object from the mirror)

Step 1: Finding the magnification
Since the convex mirror produces a 4 times enlarged image, the magnification (m) is 4.

m = -v/u = 4

Step 2: Solving for v
Using the magnification formula, we can rearrange it to solve for v:

v = -mu

Substituting the given magnification value of 4 and object distance of 8 cm:

v = -4 * 8 = -32 cm

Step 3: Finding the radius of curvature (R)
The radius of curvature (R) of a convex mirror is twice the focal length (f). So, we need to find the focal length first.

Step 3.1: Finding the focal length (f)
The focal length of a convex mirror is given by the formula:

f = R/2

Where R is the radius of curvature.

Step 3.2: Finding R
We know that the focal length (f) is half of the radius of curvature (R). Since the magnification is positive for a convex mirror, the image is virtual and the focal length is negative.

Using the mirror formula:
1/f = 1/v - 1/u

Substituting the values:
1/f = 1/-32 - 1/8
1/f = -1/32 - 4/32
1/f = -5/32

Taking the reciprocal on both sides:
f = -32/5 cm

Step 3.3: Finding R
Since f = R/2, we can rearrange it to solve for R:

R = 2f = 2 * (-32/5) = -64/5 cm

Therefore, the radius of curvature is -64/5 cm.