Question

Find out the distance of the point where the incident ray intersects the principal axis from the pole of the mirror.

• A. $\frac{15}{2}\text{cm}$
• B. $\frac{15}{8}\text{cm}$
• C. $\frac{15}{4}\text{cm}$
• D. $\frac{25}{4}\text{cm}$

Answer 1

The correct option is C $\frac{15}{4}\text{cm}$


From the figure, we can deduce that the object is at very large (theoretically infinite) distance from the pole of the concave mirror.

So, if rays are paraxial, they will pass through the focal point after reflection.

Focal distance $f=\frac{10}{2}=5\text{cm}$

Also, from the diagram, we can deduce that,
$\theta ={\cos }^{-1}\left(\frac{8}{10}\right)\Rightarrow \theta ={37}^{\circ }$

$\therefore$ Rays are not paraxial, so they are marginal.


This marginal ray after reflection passes through $\text{Q}$. Let $\text{QM}$ be the perpendicular bisector onto $\text{CN}$.

From $△CNQ$, we can say that,

$\frac{\sin \theta }{CQ}=\frac{\sin (180-2\theta )}{CN}\Rightarrow CQ=\frac{CN}{2}\sec \theta$

Using the value of $\theta ={37}^{\circ }$,

$\Rightarrow \text{CQ}=\frac{CN}{2}\sec {37}^{\circ }$

Then,
$PQ=CP-CQ=R-\frac{R}{2}\sec {37}^{\circ }=10-\frac{10}{2}\times \frac{5}{4}=\frac{15}{4}\text{cm}$

Hence, option (c) is the correct answer.

Note:- Not all the rays incident on spherical mirrors are paraxial. Condition for paraxial rays is that $\theta$ is very small, else the rays are marginal rays.