For end \(A, u_{A}=-20\, \mathrm{cm}\)

Image position of end \(A\)

\({\frac{1}{v_{A}}+\frac{1}{u_{A}}=\frac{1}{f}}\)

\(\frac{1}{{{v_A}}} + \frac{1}{{( - 20)}} = \frac{1}{{( - 10)}}\)  or  \(\frac{1}{{{v_A}}} = \frac{1}{{ - 10}} + \frac{1}{{20}} =  - \frac{1}{{20}}\)

\({v_{A}=-20\, \mathrm{cm}}\)

For end \(B, u_{B}=-30\, \mathrm{cm}\)

Image position of end \(B\) 

\(\frac{1}{v_{B}}+\frac{1}{u_{B}}=\frac{1}{f}\)

\(\frac{1}{v_{B}}+\frac{1}{(-30)}=\frac{1}{(-10)}\) or \(\frac{1}{v_{B}}=\frac{1}{-10}+\frac{1}{30}=-\frac{2}{30}\)

\(v_{B}=-15\, \mathrm{cm}\)

Length of the image

\( = \left| {{v_A}} \right| - \left| {{v_B}} \right| = \) \(20{\mkern 1mu} {\text{cm}} - 15{\mkern 1mu} {\text{cm}} = 5{\mkern 1mu} {\text{cm}}\)