Question

Two rays are incident on a spherical concave mirror of radius R = 5 cm and rays are parallel to principal axis. Choose the correct option (s)

• A. Ray 1 will intersect the principal axis at a distance of $\frac{25}{8}$ cm from pole after reflection
• B. Ray 1 will intersect the principal axis at a distance of $\frac{25}{6}$ cm from pole after reflection
• C. Ray 2 will intersect the principal axis at a distance $\frac{25}{6}$ cm from pole after reflection
• D. Ray 2 will intersect the principal axis at a distance of $\frac{25}{8}$ cm from pole after reflection

Answer 1

Answer: (A)

Ray 1 will intersect the principal axis at a distance of $\frac{25}{8}$ cm from pole after reflection

Answer 2

To determine where the reflected rays intersect the principal axis, we can follow these steps:

1. Mirror Placement: Consider the mirror as part of a circle defined by $x^2 + y^2 = 25$, with the pole at $P = (5, 0)$ and the concave side where $x \leq 5$.

2. Incidence Point for Ray 1 (h = 3 cm): The incident point $M$ is calculated as $M = (\sqrt{25 - 9}, 3) = (4, 3)$.

3. Unit Normal Vector: The unit normal vector $\hat{n}$ at point $M$ is $\hat{n} = (\frac{4}{5}, \frac{3}{5})$.

4. Reflected Direction Vector: For an incident ray with direction $(-1, 0)$, the reflected direction $v_r$ is given by:

   $v_r = v_{in} - 2(v_{in} \cdot \hat{n})\hat{n} = (\frac{7}{25}, \frac{24}{25})$.

5. Intersection with Principal Axis: By setting $y = 0$ for the line passing through $M$ with direction $v_r$, the intersection point $x$ is found to be $\frac{25}{8}$ cm from $P$.

A similar calculation for the second ray ($h = -4$ cm) will yield a different intersection point, confirming that only ray 1 intersects the axis at $\frac{25}{8}$ cm.