A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle `theta``=30degree` to the axis of the lens, as shown in the figure
If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point `(x,y)` at which the image is formed are

Options

A. `(25,25sqrt(3))`
B. (0, 0)
C. `(125//3, 25sqrt(3))`
D. `(50 -25// sqrt(3), 25)`

Answer

A

Explanation

First let us find the location of image formed by convex lens using lens formula.
`(1)/(f)=(1)/(v)-(1)/(u)`
`rArr" "(1)/(+30)=(1)/(v)-(1)/(-50)`
On solving we get `v = 75 cm`.
Let us first assume that mirror is not inclined rather assume its axis coincides with that of lens. In that case image formed by convex lens will be 25 cm inside the mirror and hence it will act as virtual object for convex mirror, Let us apply mirror formula to get the location of image formed by convex mirror.
`(1)/(f)=(1)/(v)+(1)/(u)`
Radius of curvature of mirror is 100 cm and hence its focal length is 50 cm.
`rArr" "(1)/(+50)=(1)/(v)+(1)/(+25)`
On solving we get `v = -50 cm`. Hence when mirror is not inclined then final image is formed at the centre of lens and hence at (0,0) but when mirror is rotated by an angle `30^(@)` then reflected ray gets rotated by an angle `60^(@)`.

Now we can see in figure that
`a=50cos 60^(@)=25`
`b=50sin 60^(@)=25sqrt3`
Coordinates of final location of image are `(50-a,b)`
Hence `(25, 25sqrt3)`
Hence, option (a) is correct.