The magnification \( m \) for a concave mirror is given by: \[ m = \frac{-v}{u} \] where \( v \) is the image distance and \( u \) is the object distance. We are given that \( m = -\frac{1}{9} \) for a real image, so: \[ \frac{-v}{u} = -\frac{1}{9} \] This implies: \[ v = \frac{u}{9} \] The mirror equation is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] For a concave mirror, the focal length \( f \) is related to the radius of curvature \( R \) by: \[ f = \frac{R}{2} = \frac{36}{2} = 18 \, \text{cm} \] Substitute \( f = 18 \) cm into the mirror equation: \[ \frac{1}{18} = \frac{1}{v} + \frac{1}{u} \] Using the relation \( v = \frac{u}{9} \), substitute this into the equation: \[ \frac{1}{18} = \frac{9}{u} + \frac{1}{u} \] Simplifying: \[ \frac{1}{18} = \frac{10}{u} \] Solving for \( u \): \[ u = 180 \, \text{cm} \] Thus, the object should be placed 180 cm from the mirror.
Was this answer helpful?
2
0
Show Solution
Verified By Collegedunia
Approach Solution -2
Using the New Cartesian sign convention, distances measured against the incoming light are negative. Let the object distance be \(u = -x\) (so \(x\) is the magnitude we want), the focal length of the concave mirror be \(f = -\dfrac{R}{2} = -18\,\text{cm}\), and the magnification for a real image one-ninth the object's size be \(m = -\dfrac{1}{9}\) (negative because real images from a concave mirror are inverted). Since \(m = -v/u\), we get \(v = \dfrac{u}{9} = -\dfrac{x}{9}\). Substituting into the mirror formula \( \dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f} \) gives \( -\dfrac{9}{x} - \dfrac{1}{x} = -\dfrac{1}{18} \), so \( \dfrac{10}{x} = \dfrac{1}{18} \), which gives \( x = 180\,\text{cm} \). Let's test the listed distances against this same equation.
170 cm: Putting \(x = 170\) into \(10/x\) gives about \(0.0588\), while \(1/18 \approx 0.0556\); these are not equal, so 170 cm does not satisfy the mirror equation with this magnification.
180 cm: Putting \(x = 180\) gives \(10/180 = 1/18\) exactly, matching the focal-length term precisely, so this distance satisfies both the mirror formula and the magnification condition.
190 cm: Putting \(x = 190\) gives \(10/190 \approx 0.0526\), which is smaller than \(1/18 \approx 0.0556\), so it does not balance the equation either.
None: Since 180 cm does satisfy the equation exactly, this option is not needed.
Only 180 cm makes the mirror equation and the one-ninth magnification condition hold together at the same time.
