# A concave mirror (f = 36 \ cm) produces an image whose distance from the mirror is one-third the...

## Question:

A concave mirror {eq}(f = 36 \ cm) {/eq} produces an image whose distance from the mirror is one-third the object distance.

Determine:

a) the object distance and

b) the (positive) image distance.

## Spherical mirror

A spherical mirror can be classified into two categories. First, one whose striking surface within the sphere, known as a concave mirror. The second one, whose striking surface does not lie inside that is outside the sphere, is termed as a convex mirror.

Given data

• The focal length of the concave mirror is: {eq}f = - 36\;{\rm{cm}} {/eq}
• The image distance is: {eq}v = \dfrac{u}{3} {/eq}

The expression for the mirror formula is,

{eq}\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f} {/eq}

Here, {eq}u {/eq} is the object distance.

Substitute the given value in the above expression.

{eq}\begin{align*} \dfrac{3}{u} + \dfrac{1}{u} &= - \dfrac{1}{{36}}\\ \dfrac{4}{u} &= \dfrac{1}{{36}}\\ u &= - 144\;{\rm{cm}} \end{align*} {/eq}

Substitute the given value of the object distance into the expression of the image distance.

{eq}\begin{align*} v &= - \dfrac{{144\;{\rm{m}}}}{3}\\ &= - 48\;{\rm{cm}} \end{align*} {/eq}

Thus, the object distance is {eq}144\;{\rm{cm}} {/eq} in front of the mirror and image distance is {eq}48\;{\rm{cm}} {/eq} in front of mirror.