A luminous point A is inside a circular track. A ray emerges from A and after two reflections at the circle returns to A. If $\alpha$ be the angle of incidence, x the distance from the centre of the circle to A and Y the distance of the centre from the point D, where the ray in its course crosses the diameter through A, prove that tan $\alpha$=$\sqrt{\frac{\left(x-y\right)}{\left(x+y\right)}}.$

A concave mirror produces a real image twice the size of the object when placed at a distance of 22.5 cm from it. Where should the object be place so that the size of the image becomes three times the size of the object ?

A plane glass plate is kept between a point object and a convex mirror of focal length 20 cm. The distance of the glass plate from the mirror is 5 cm. Rays of light from the point source, after reflection at the convex mirror and also at the front face of the glass plate form images at the same position. Find the distance between the glass plate and the point source.

An object is kept at a distance of 40 cm from a convex mirror. At a distance of 25 cm from the object a plane mirror is so placed that it covers one half of the convex mirror. As a result, the two images produced by the curvature of the convex mirror ?

A point source S is placed midway between two converging mirrors having equal focal length f. Find the distance between the mirrors for which only one image is formed.

An object 2.5 cm long is placed vertically on the principal axis in front of a concave mirror of focal length f its distance $\frac{3}{4}f$ from it. Of what length and nature are its images formed?