The radius of curvature of a concave mirror is 40 cm. If a real image twice the size of the object is formed, the object distance is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 30 cm
A concave mirror has \( f=R/2 \), negative in the Cartesian sign convention. A real image is inverted, so the magnification \( m=-v/u \) is negative.
\( \begin{aligned} f &= -\frac{R}{2} \\ &= -\frac{40\,\mathrm{cm}}{2} \\ &= -20\,\mathrm{cm} \end{aligned} \)
The real image is twice the size of the object, so \( m=-2 \):
\( \displaystyle -\frac{v}{u}=-2 \)
\( \Rightarrow\; v=2u \)
Putting this into the mirror equation:
\( \displaystyle \frac{1}{v}+\frac{1}{u}=\frac{1}{f} \)
\( \displaystyle \frac{1}{2u}+\frac{1}{u}=-\frac{1}{20\,\mathrm{cm}} \)
\( \displaystyle \frac{3}{2u}=-\frac{1}{20\,\mathrm{cm}} \)
\( \Rightarrow\; u=-30\,\mathrm{cm} \)
The minus sign places the object in front of the mirror. Therefore, the object distance is 30 cm.