A thin rod lies along the principal axis of a concave mirror of focal length 30 cm . The images of the two ends of the rod have transverse magnifications of 3 (real image) and 2 (virtual image) respectively. The length of the image of the rod is ______ cm .

Step-by-step solution

Let the coordinates of the two ends of the rod be u_A and u_B . The focal length of the concave mirror is f = -30 cm . For the end forming a real image with transverse magnification 3 , the magnification is m_A = -3 . Using the formula m = f f - u : -3 = -30 -30 - u_A -30 - u_A = 10 u_A = -40 cm The image coordinate v_A is given by v_A = -m_A u_A = -(-3)(-40) = -120 cm . For the end forming a virtual image with transverse magnification 2 , the magnification is m_B = +2 . 2 = -30 -30 - u_B -30 - u_B = -15 u_B = -15