The correct answer is $\boxed{\frac{3}{4}f}$.
The focal length of a mirror is the distance from the mirror to the point where all light rays parallel to the principal axis converge after reflection. The focal length of a mirror depends on the refractive index of the medium in which it is placed. The refractive index of a medium is a measure of how much light slows down when it passes through the medium. The refractive index of air is 1, and the refractive index of water is 1.33.
When a convex mirror is immersed in a liquid, the focal length of the mirror decreases. This is because the light rays are refracted as they pass from air to the liquid. The amount of refraction depends on the difference in the refractive indices of the two media.
The greater the difference in the refractive indices, the greater the refraction.In this case, the refractive index of water is greater than the refractive index of air. This means that the light rays will be refracted
towards the normal when they pass from air to water. This will cause the focal length of the mirror to decrease.The focal length of the mirror in the liquid can be calculated using the following formula:
$f_l = \frac{f_a}{\mu_a – \mu_l}$
where $f_l$ is the focal length of the mirror in the liquid, $f_a$ is the focal length of the mirror in air, $\mu_a$ is the refractive index of air, and $\mu_l$ is the refractive index of the liquid.
In this case, $f_a = f$, $\mu_a = 1$, and $\mu_l = \frac{4}{3}$. Substituting these values into the formula gives:
$f_l = \frac{f}{\frac{1}{\frac{4}{3}} – 1} = \frac{3}{4}f$
Therefore, the focal length of the mirror in the liquid is $\frac{3}{4}f$.