Distance between object and its image \(\left( \text{magnified by } -\frac{1}{3}\right)\) is 30 cm. The focal length of the mirror used is \(\left(\frac{x}{4}\right) \mathrm{cm}\),
where magnitude of value of \(x\) is _________.

\(\begin{aligned} & M=-\frac{1}{3} \\ & -\frac{-V}{-U}=\frac{-1}{3} \Rightarrow V=\frac{U}{3}\end{aligned}\)
Distance \(b / w\) object and image :

\(\begin{aligned} & \mathrm{U}-\mathrm{V}=30 \\ & \mathrm{U}-\frac{\mu}{3}=30 \\ & \Rightarrow \mathrm{U}=45 \quad \mathrm{~V}=15 \\ & \frac{1}{\mathrm{f}}=\frac{1}{\mathrm{~V}}+\frac{1}{\mathrm{U}}=-\frac{1}{15}-\frac{1}{45} \\ & \Rightarrow \mathrm{~F}=\frac{45}{4} \\ & \mathrm{x}=45\end{aligned}\)