Read the following passage and answer the questions.
Riya uses a concave make-up mirror of radius of curvature 60 cm. She holds her face 10 cm in front of it. In the morning the mirror stands on a sunny window sill. A narrow parallel beam of sunlight then falls on it at a small angle to its principal axis. Her father’s car has a convex side-view mirror of radius of curvature 2 m. Sitting in the parked car, he watches another car come up from behind at a steady speed. He looks at it in the mirror when it is 19 m away and again when it is 4 m away. Both mirrors are small compared with their radii of curvature, so all the rays are paraxial. All distances are measured from the pole of each mirror, using the Cartesian sign convention.
The ratio of the height of the image of the car at 4 m to the height of its image at 19 m is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4
A convex mirror has its focus behind the mirror, so its focal length is positive. With \( R=2\ \mathrm{m} \), \( f=R/2 \) gives \( f=+1\ \mathrm{m} \). The car is in front of the mirror, so \( u \) is negative. The mirror equation gives \( v \) for each position, and \( m=-v/u \) gives the size of the image.
When the car is 19 m away:
\( u_1=-19\ \mathrm{m} \)
\( \begin{aligned} \frac{1}{v_1} &= \frac{1}{f}-\frac{1}{u_1} \\ &= 1+\frac{1}{19} \\ &= \frac{20}{19} \end{aligned} \)
\( v_1=0.95\ \mathrm{m} \)
\( \begin{aligned} m_1 &= -\frac{v_1}{u_1} \\ &= \frac{0.95}{19} \\ &= \frac{1}{20} \end{aligned} \)
When the car is 4 m away:
\( u_2=-4\ \mathrm{m} \)
\( \begin{aligned} \frac{1}{v_2} &= \frac{1}{f}-\frac{1}{u_2} \\ &= 1+\frac{1}{4} \\ &= \frac{5}{4} \end{aligned} \)
\( v_2=0.8\ \mathrm{m} \)
\( \begin{aligned} m_2 &= -\frac{v_2}{u_2} \\ &= \frac{0.8}{4} \\ &= \frac{1}{5} \end{aligned} \)
The ratio of the image heights is the ratio of the magnifications:
\( \begin{aligned} \frac{h_2^{\prime}}{h_1^{\prime}} &= \frac{m_2}{m_1} \\ &= \frac{1/5}{1/20} \\ &= 4 \end{aligned} \)
Hence, the image of the car at 4 m is 4 times as tall as its image at 19 m.
NCERT (2026-27):
Focal length of spherical mirrors – “… Now, for small θ, the point D is very close to the point P. Therefore, FD = f and CD = R. …”
\( \displaystyle f=R/2 \)\( \qquad (9.3) \)
The mirror equation – “Dividing it by v, we get”
\( \displaystyle \frac{1}{v}+\frac{1}{u}=\frac{1}{f} \)\( \qquad (9.7) \)
The mirror equation – “… so that”
\( \displaystyle m=\frac{h^{\prime}}{h}=-\frac{v}{u} \)\( \qquad (9.9) \)