An object kept in front of a spherical mirror of focal length 25 cm produces an erect and virtual image five times the size of object.  Find the object distance along with sign.

Sure, let's solve the problem step by step.

Given: Focal length (f) = 25 cm Image height (h') = 5 times the object height (h)

Step 1: Determine the type of mirror Since the image formed is erect and virtual, it indicates that the mirror is a concave mirror.

Step 2: Use the mirror formula The mirror formula is given by: 1/f = 1/v - 1/u

Where: f = focal length of the mirror v = image distance from the mirror u = object distance from the mirror

Step 3: Determine the magnification (m) The magnification (m) is given by: m = -v/u

Since the image is 5 times the size of the object, we have: m = h'/h = 5

Step 4: Substitute the given values into the mirror formula 1/f = 1/v - 1/u

Substituting the values: 1/25 = 1/v - 1/u

Step 5: Substitute the magnification value into the magnification formula m = -v/u

Substituting the values: 5 = -v/u

Step 6: Solve the equations simultaneously From step 4, we have: 1/25 = 1/v - 1/u

From step 5, we have: 5 = -v/u

Rearranging the equations, we get: 1/v = 1/25 + 1/u v/u = -5

Step 7: Substitute the value of v/u into the equation 1/v = 1/25 + 1/u -5 = 1/25 + 1/u

Simplifying the equation, we get: -5 = (1 + 25)/25u

-5 = 26/25u

Cross-multiplying, we get: -5 * 25u = 26

-125u = 26

Dividing both sides by -125, we get: u = -26/125

Therefore, the object distance (u) is -26/125 cm.

Note: The negative sign indicates that the object is placed in front of the mirror.

So, the object distance is -26/125 cm (with a negative sign).